{"id":112,"date":"2025-04-09T17:10:29","date_gmt":"2025-04-09T17:10:29","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/2-6-other-types-of-equations-college-algebra-2e-openstax\/"},"modified":"2026-03-19T15:56:33","modified_gmt":"2026-03-19T15:56:33","slug":"2-6-other-types-of-equations","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/2-6-other-types-of-equations\/","title":{"raw":"2.6 Other Types of Equations","rendered":"2.6 Other Types of Equations"},"content":{"raw":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\r\n<div id=\"page_b88d40b3-a0d5-4a8f-8b2e-85993b911cc0\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\r\n<div class=\"ui-has-child-title\" data-type=\"abstract\"><header>\r\n<h2 data-type=\"title\">Learning Objectives<\/h2>\r\n<\/header><section>\r\n<p id=\"para-00001\">In this section, you will:<\/p>\r\n\r\n<ul id=\"list-00001\">\r\n \t<li>Solve equations involving rational exponents.<\/li>\r\n \t<li>Solve equations using factoring.<\/li>\r\n \t<li>Solve radical equations.<\/li>\r\n \t<li>Solve absolute value equations.<\/li>\r\n \t<li>Solve other types of equations.<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<p id=\"fs-id1519532\">We have solved linear equations, rational equations, and quadratic equations using several methods. However, there are many other types of equations, and we will investigate a few more types in this section. We will look at equations involving rational exponents, polynomial equations, radical equations, absolute value equations, equations in quadratic form, and some rational equations that can be transformed into quadratics. Solving any equation, however, employs the same basic algebraic rules. We will learn some new techniques as they apply to certain equations, but the algebra never changes.<\/p>\r\n\r\n<section id=\"fs-id2485804\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Solving Equations Involving Rational Exponents<\/h2>\r\n<p id=\"fs-id1274976\">Rational exponents are exponents that are fractions, where the numerator is a power and the denominator is a root. For example,<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msup>\r\n<mrow>\r\n<mn>16<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\r\n<mrow>\r\n<mn>16<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is another way of writing\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msqrt>\r\n<mn>16<\/mn>\r\n<\/msqrt>\r\n<mo>;<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msqrt>\r\n<mn>16<\/mn>\r\n<\/msqrt>\r\n<mo>;<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mn>8<\/mn>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mn>8<\/mn>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is another way of writing\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mtext>\u200b<\/mtext><mspace width=\"0.5em\"><\/mspace><mroot>\r\n<mn>8<\/mn>\r\n<mn>3<\/mn>\r\n<\/mroot>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtext>\u200b<\/mtext><mspace width=\"0.5em\"><\/mspace><mroot>\r\n<mn>8<\/mn>\r\n<mn>3<\/mn>\r\n<\/mroot>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>The ability to work with rational exponents is a useful skill, as it is highly applicable in calculus.\r\n<p id=\"fs-id1723049\">We can solve equations in which a variable is raised to a rational exponent by raising both sides of the equation to the reciprocal of the exponent. The reason we raise the equation to the reciprocal of the exponent is because we want to eliminate the exponent on the variable term, and a number multiplied by its reciprocal equals 1. For example,<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and so on.\r\n<div id=\"fs-id1319605\" class=\"ui-has-child-title\" data-type=\"note\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Rational Exponents<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1532274\">A rational exponent indicates a power in the numerator and a root in the denominator. There are multiple ways of writing an expression, a variable, or a number with a rational exponent:<\/p>\r\n\r\n<div id=\"fs-id1512276\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mi>m<\/mi>\r\n<mi>n<\/mi>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mi>n<\/mi>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mi>m<\/mi>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mi>m<\/mi>\r\n<\/msup>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mi>n<\/mi>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><mroot>\r\n<mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mi>m<\/mi>\r\n<\/msup>\r\n<\/mrow>\r\n<mi>n<\/mi>\r\n<\/mroot>\r\n<mo>=<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mroot>\r\n<mi>a<\/mi>\r\n<mi>n<\/mi>\r\n<\/mroot>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mi>m<\/mi>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mi>m<\/mi>\r\n<mi>n<\/mi>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mi>n<\/mi>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mi>m<\/mi>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mi>m<\/mi>\r\n<\/msup>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mi>n<\/mi>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><mroot>\r\n<mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mi>m<\/mi>\r\n<\/msup>\r\n<\/mrow>\r\n<mi>n<\/mi>\r\n<\/mroot>\r\n<mo>=<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mroot>\r\n<mi>a<\/mi>\r\n<mi>n<\/mi>\r\n<\/mroot>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mi>m<\/mi>\r\n<\/msup>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_06_01\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">1<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1458449\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2592723\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Evaluating a Number Raised to a Rational Exponent<\/h3>\r\n<p id=\"fs-id1239738\">Evaluate<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msup>\r\n<mn>8<\/mn>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\r\n<mn>8<\/mn>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id2522284\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id2422284\">Whether we take the root first or the power first depends on the number. It is easy to find the cube root of 8, so rewrite<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msup>\r\n<mn>8<\/mn>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\r\n<mn>8<\/mn>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>as\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mn>8<\/mn>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mn>8<\/mn>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id1227998\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mn>8<\/mn>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>4<\/mn><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mn>8<\/mn>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>4<\/mn><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1514351\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#1<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_06_01\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1336704\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2400251\">Evaluate<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msup>\r\n<mrow>\r\n<mn>64<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\r\n<mrow>\r\n<mn>64<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_06_02\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">2<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id2385021\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3264211\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solve the Equation Including a Variable Raised to a Rational Exponent<\/h3>\r\n<p id=\"fs-id1198214\">Solve the equation in which a variable is raised to a rational exponent:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>5<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><mn>32.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>5<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><mn>32.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id2431600\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1569151\">The way to remove the exponent on <em data-effect=\"italics\">x<\/em> is by raising both sides of the equation to a power that is the reciprocal of<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mfrac>\r\n<mn>5<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mfrac>\r\n<mn>5<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>which is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mfrac>\r\n<mn>4<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mfrac>\r\n<mn>4<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id1277511\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>5<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>32<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>5<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>4<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>32<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>4<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>The\u00a0fifth\u00a0root\u00a0of\u00a032\u00a0is\u00a02.<\/mtext><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>16<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>5<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>32<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>5<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>4<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>32<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>4<\/mn>\r\n<mn>5<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>The\u00a0fifth\u00a0root\u00a0of\u00a032\u00a0is\u00a02.<\/mtext><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>16<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1759744\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#2<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_06_02\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2293541\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1314798\">Solve the equation<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><mn>125.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><mn>125.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_06_03\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">3<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id2437878\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1335480\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving an Equation Involving Rational Exponents and Factoring<\/h3>\r\n<p id=\"fs-id1269017\">Solve<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1522050\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1296613\">This equation involves rational exponents as well as factoring rational exponents. Let us take this one step at a time. First, put the variable terms on one side of the equal sign and set the equation equal to zero.<\/p>\r\n\r\n<div id=\"fs-id1400272\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>0<\/mn><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>0<\/mn><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1368636\">Now, it looks like we should factor the left side, but what do we factor out? We can always factor the term with the lowest exponent. Rewrite<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>as\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>Then, factor out\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>from both terms on the left.\r\n<div id=\"fs-id1441452\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1933337\">Where did<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>come from? Remember, when we multiply two numbers with the same base, we add the exponents. Therefore, if we multiply\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>back in using the distributive property, we get the expression we had before the factoring, which is what should happen. We need an exponent such that when added to\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mfrac>\r\n<mn>2<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mfrac>\r\n<mn>2<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>equals\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>Thus, the exponent on <em data-effect=\"italics\">x <\/em>in the parentheses is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<p id=\"fs-id1891109\">Let us continue. Now we have two factors and can use the zero factor theorem.<\/p>\r\n\r\n<div id=\"fs-id1422353\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>1<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Divide\u00a0both\u00a0sides\u00a0by\u00a03<\/mtext><mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Raise\u00a0both\u00a0sides\u00a0to\u00a0the\u00a0reciprocal\u00a0of\u00a0<\/mtext><mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mn>81<\/mn><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>1<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Divide\u00a0both\u00a0sides\u00a0by\u00a03<\/mtext><mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Raise\u00a0both\u00a0sides\u00a0to\u00a0the\u00a0reciprocal\u00a0of\u00a0<\/mtext><mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mn>81<\/mn><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2957928\">The two solutions are<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>0<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mn>81<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<mn>81<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1267637\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#3<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_06_03\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1411367\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1441991\">Solve:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><mn>8.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><mn>8.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2945385\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Solving Equations Using Factoring<\/h2>\r\n<p id=\"fs-id2521696\">We have used factoring to solve quadratic equations, but it is a technique that we can use with many types of polynomial equations, which are equations that contain a string of terms including numerical coefficients and variables. When we are faced with an equation containing polynomials of degree higher than 2, we can often solve them by factoring.<\/p>\r\n\r\n<div id=\"fs-id1786047\" class=\"ui-has-child-title\" data-type=\"note\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Polynomial Equations<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id3231810\">A polynomial of degree <em data-effect=\"italics\">n <\/em>is an expression of the type<\/p>\r\n\r\n<div id=\"fs-id2905918\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<msub>\r\n<mi>a<\/mi>\r\n<mi>n<\/mi>\r\n<\/msub>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mi>n<\/mi>\r\n<\/msup>\r\n<mo>+<\/mo><msub>\r\n<mi>a<\/mi>\r\n<mrow>\r\n<mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/msub>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>+<\/mo><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>\u22c5<\/mo><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>\u22c5<\/mo><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>\u22c5<\/mo><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>+<\/mo><msub>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><msub>\r\n<mi>a<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mi>x<\/mi><mo>+<\/mo><msub>\r\n<mi>a<\/mi>\r\n<mn>0<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msub>\r\n<mi>a<\/mi>\r\n<mi>n<\/mi>\r\n<\/msub>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mi>n<\/mi>\r\n<\/msup>\r\n<mo>+<\/mo><msub>\r\n<mi>a<\/mi>\r\n<mrow>\r\n<mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/msub>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>+<\/mo><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>\u22c5<\/mo><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>\u22c5<\/mo><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>\u22c5<\/mo><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>+<\/mo><msub>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><msub>\r\n<mi>a<\/mi>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mi>x<\/mi><mo>+<\/mo><msub>\r\n<mi>a<\/mi>\r\n<mn>0<\/mn>\r\n<\/msub>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2444064\">where <em data-effect=\"italics\">n<\/em> is a positive integer and<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msub>\r\n<mi>a<\/mi>\r\n<mi>n<\/mi>\r\n<\/msub>\r\n<mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub>\r\n<mi>a<\/mi>\r\n<mn>0<\/mn>\r\n<\/msub>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msub>\r\n<mi>a<\/mi>\r\n<mi>n<\/mi>\r\n<\/msub>\r\n<mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub>\r\n<mi>a<\/mi>\r\n<mn>0<\/mn>\r\n<\/msub>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>are real numbers and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msub>\r\n<mi>a<\/mi>\r\n<mi>n<\/mi>\r\n<\/msub>\r\n<mo>\u2260<\/mo><mn>0.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msub>\r\n<mi>a<\/mi>\r\n<mi>n<\/mi>\r\n<\/msub>\r\n<mo>\u2260<\/mo><mn>0.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<p id=\"fs-id2931289\">Setting the polynomial equal to zero gives a <span id=\"term-00001\" data-type=\"term\">polynomial equation<\/span>. The total number of solutions (real and complex) to a polynomial equation is equal to the highest exponent <em data-effect=\"italics\">n<\/em>.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_06_04\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">4<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1996696\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2501149\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving a Polynomial by Factoring<\/h3>\r\n<p id=\"fs-id1391817\">Solve the polynomial by factoring:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>5<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>80<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>5<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>80<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1517483\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1419163\">First, set the equation equal to zero. Then factor out what is common to both terms, the GCF.<\/p>\r\n\r\n<div id=\"fs-id2502503\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>5<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>80<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>5<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>(<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>16<\/mn><mo>)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>5<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>80<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>5<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>(<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>16<\/mn><mo>)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1924982\">Notice that we have the difference of squares in the factor<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>16<\/mn><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>16<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>which we will continue to factor and obtain two solutions. The first term,\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>5<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>5<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>generates, technically, two solutions as the exponent is 2, but they are the same solution.\r\n<div id=\"fs-id2824827\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>5<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>16<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>4<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22124<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>5<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>16<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>4<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22124<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2638784\">The solutions are<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>0<\/mn><mspace width=\"0.5em\"><\/mspace><mtext>(double\u00a0solution),<\/mtext>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>0<\/mn><mspace width=\"0.5em\"><\/mspace><mtext>(double\u00a0solution),<\/mtext>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>4<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>4<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>\u22124.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22124.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/details>\r\n<div id=\"fs-id1693367\" data-type=\"commentary\">\r\n<h3 data-type=\"commentary-title\"><span class=\"os-title-label\">Analysis <\/span><\/h3>\r\n<p id=\"fs-id768264\">We can see the solutions on the graph in <a class=\"autogenerated-content\" href=\"2-6-other-types-of-equations#Figure_02_06_001\">Figure 1<\/a>. The <em data-effect=\"italics\">x-<\/em>coordinates of the points where the graph crosses the <em data-effect=\"italics\">x-<\/em>axis are the solutions\u2014the <em data-effect=\"italics\">x-<\/em>intercepts. Notice on the graph that at the solution<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>the graph touches the <em data-effect=\"italics\">x-<\/em>axis and bounces back. It does not cross the <em data-effect=\"italics\">x-<\/em>axis. This is typical of double solutions.\r\n<div id=\"Figure_02_06_001\" class=\"os-figure\">\r\n<figure class=\"small\" data-id=\"Figure_02_06_001\"><span id=\"fs-id1548359\" data-type=\"media\" data-alt=\"Coordinate plane with the x-axis ranging from negative 5 to 5 and the y-axis ranging from negative 400 to 500 in intervals of 100. The function five times x to the fourth power minus eighty x squared equals zero is graphed along with the points (negative 4,0), (0,0), and (4,0).\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Coordinate plane with the x-axis ranging from negative 5 to 5 and the y-axis ranging from negative 400 to 500 in intervals of 100. The function five times x to the fourth power minus eighty x squared equals zero is graphed along with the points (negative 4,0), (0,0), and (4,0).\" width=\"487\" height=\"401\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/2e49a8dfcd752028a5ab6adfd4ad7bfdcb6e68a4\" \/>\r\n<\/span><\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span>\r\n<span class=\"os-number\">1<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id988839\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#4<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_06_04\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2496351\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3040346\">Solve by factoring:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>12<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>12<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_06_05\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">5<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id2435055\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1007381\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solve a Polynomial by Grouping<\/h3>\r\n<p id=\"fs-id2501656\">Solve a polynomial by grouping:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msup>\r\n<mi>x<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>9<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>9<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\r\n<mi>x<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>9<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>9<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1475994\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1422651\">This polynomial consists of 4 terms, which we can solve by grouping. Grouping procedures require factoring the first two terms and then factoring the last two terms. If the factors in the parentheses are identical, we can continue the process and solve, unless more factoring is suggested.<\/p>\r\n\r\n<div id=\"fs-id2516826\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>9<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>9<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>9<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mo>(<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>9<\/mn><mo>)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>9<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>9<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>9<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mo>(<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>9<\/mn><mo>)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1408391\">The grouping process ends here, as we can factor<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>9<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>9<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>\r\nusing the difference of squares formula.\r\n<div id=\"fs-id1402146\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd>\r\n<mrow>\r\n<mo>(<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>9<\/mn><mo>)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>3<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22123<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22121<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd>\r\n<mrow>\r\n<mo>(<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>9<\/mn><mo>)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>3<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22123<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22121<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2528732\">The solutions are<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22123<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22123<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>\u22121.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22121.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>Note that the highest exponent is 3 and we obtained 3 solutions. We can see the solutions, the <em data-effect=\"italics\">x-<\/em>intercepts, on the graph in <a class=\"autogenerated-content\" href=\"2-6-other-types-of-equations#Figure_02_06_002\">Figure 2<\/a><strong>.<\/strong>\r\n<div id=\"Figure_02_06_002\" class=\"os-figure\">\r\n<figure class=\"small\" data-id=\"Figure_02_06_002\"><span id=\"fs-id2304259\" data-type=\"media\" data-alt=\"Coordinate plane with the x-axis ranging from negative 5 to 5 and the y-axis ranging from negative 30 to 20 in intervals of 5. The function x cubed plus x squared minus nine times x minus nine equals zero is graphed along with the points (negative 3,0), (negative 1,0), and (3,0).\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Coordinate plane with the x-axis ranging from negative 5 to 5 and the y-axis ranging from negative 30 to 20 in intervals of 5. The function x cubed plus x squared minus nine times x minus nine equals zero is graphed along with the points (negative 3,0), (negative 1,0), and (3,0).\" width=\"402\" height=\"370\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/94b4e297364251745bf2d31918af414b117764c9\" \/>\r\n<\/span><\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span>\r\n<span class=\"os-number\">2<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/details>\r\n<div id=\"fs-id2504109\" data-type=\"commentary\">\r\n<h3 data-type=\"commentary-title\"><span class=\"os-title-label\">Analysis <\/span><\/h3>\r\n<p id=\"fs-id1243628\">We looked at solving quadratic equations by factoring when the leading coefficient is 1. When the leading coefficient is not 1, we solved by grouping. Grouping requires four terms, which we obtained by splitting the linear term of quadratic equations. We can also use grouping for some polynomials of degree higher than 2, as we saw here, since there were already four terms.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2506398\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Solving Radical Equations<\/h2>\r\n<p id=\"fs-id1846458\"><strong>Radical equations<\/strong> are equations that contain variables in the <span id=\"term-00002\" class=\"no-emphasis\" data-type=\"term\">radicand<\/span> (the expression under a radical symbol), such as<\/p>\r\n\r\n<div id=\"fs-id2781660\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>18<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mo>\u2212<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>2<\/mn>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>18<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mo>\u2212<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>2<\/mn>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1719179\">Radical equations may have one or more radical terms, and are solved by eliminating each radical, one at a time. We have to be careful when solving radical equations, as it is not unusual to find <span id=\"term-00003\" data-type=\"term\">extraneous solutions<\/span>, roots that are not, in fact, solutions to the equation. These solutions are not due to a mistake in the solving method, but result from the process of raising both sides of an equation to a power. However, checking each answer in the original equation will confirm the true solutions.<\/p>\r\n\r\n<div id=\"fs-id2506978\" class=\"ui-has-child-title\" data-type=\"note\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Radical Equations<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"eip-id3052762\">An equation containing terms with a variable in the radicand is called a <span id=\"term-00004\" data-type=\"term\">radical equation<\/span>.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2307719\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1517012\"><strong>Given a radical equation, solve it.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id3143660\" type=\"1\">\r\n \t<li>Isolate the radical expression on one side of the equal sign. Put all remaining terms on the other side.<\/li>\r\n \t<li>If the radical is a square root, then square both sides of the equation. If it is a cube root, then raise both sides of the equation to the third power. In other words, for an <em data-effect=\"italics\">n<\/em>th root radical, raise both sides to the <em data-effect=\"italics\">n<\/em>th power. Doing so eliminates the radical symbol.<\/li>\r\n \t<li>Solve the remaining equation.<\/li>\r\n \t<li>If a radical term still remains, repeat steps 1\u20132.<\/li>\r\n \t<li>Confirm solutions by substituting them into the original equation.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_06_06\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">6<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1222816\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1551039\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving an Equation with One Radical<\/h3>\r\n<p id=\"fs-id1520471\">Solve<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msqrt>\r\n<mrow>\r\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mi>x<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msqrt>\r\n<mrow>\r\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mi>x<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id2381217\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1332305\">The radical is already isolated on the left side of the equal side, so proceed to square both sides.<\/p>\r\n\r\n<div id=\"fs-id1919300\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1895386\">We see that the remaining equation is a quadratic. Set it equal to zero and solve.<\/p>\r\n\r\n<div id=\"fs-id1905278\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22125<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>3<\/mn>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22125<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>3<\/mn>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1257720\">The proposed solutions are<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22125<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22125<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>Let us check each solution back in the original equation. First, check\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>x<\/mi><mo>=<\/mo><mn>\u22125.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>=<\/mo><mn>\u22125.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id1463741\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22125<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>25<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22125<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mn>5<\/mn>\r\n<\/mtd>\r\n<mtd>\r\n<mo>\u2260<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo><\/mo><mn>\u22125<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22125<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>25<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22125<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mn>5<\/mn>\r\n<\/mtd>\r\n<mtd>\r\n<mo>\u2260<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo><\/mo><mn>\u22125<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2628354\">This is an extraneous solution. While no mistake was made solving the equation, we found a solution that does not satisfy the original equation.<\/p>\r\n<p id=\"fs-id1417225\">Check<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>x<\/mi><mo>=<\/mo><mn>3.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>=<\/mo><mn>3.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id1515949\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>3<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mn>9<\/mn>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>3<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mn>3<\/mn>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>3<\/mn>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>3<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mn>9<\/mn>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>3<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mn>3<\/mn>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>3<\/mn>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1199291\">The solution is<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>3.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>3.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1441693\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#5<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_06_05\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1583879\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1583880\">Solve the radical equation:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_06_07\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">7<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1720874\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2499586\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving a Radical Equation Containing Two Radicals<\/h3>\r\n<p id=\"fs-id1410869\">Solve<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>+<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mn>4.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>+<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mn>4.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1719019\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1929508\">As this equation contains two radicals, we isolate one radical, eliminate it, and then isolate the second radical.<\/p>\r\n\r\n<div id=\"fs-id2268062\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mo>+<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>4<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Subtract\u00a0<\/mtext><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mspace width=\"0.5em\"><\/mspace><mtext>from\u00a0both\u00a0sides<\/mtext><mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo>(<\/mo><mn>4<\/mn><mo>\u2212<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mo>)<\/mo><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Square\u00a0both\u00a0sides<\/mtext><mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mo>+<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>4<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Subtract\u00a0<\/mtext><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mspace width=\"0.5em\"><\/mspace><mtext>from\u00a0both\u00a0sides<\/mtext><mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo>(<\/mo><mn>4<\/mn><mo>\u2212<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mo>)<\/mo><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Square\u00a0both\u00a0sides<\/mtext><mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1548786\">Use the perfect square formula to expand the right side:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mn>\u22122<\/mn><mi>a<\/mi><mi>b<\/mi><mo>+<\/mo><msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mn>\u22122<\/mn><mi>a<\/mi><mi>b<\/mi><mo>+<\/mo><msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id2506054\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>+<\/mo><msup>\r\n<mrow>\r\n<mo>(<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>)<\/mo>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>16<\/mn><mo>\u2212<\/mo><mn>8<\/mn><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>14<\/mn><mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>8<\/mn><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Combine\u00a0like\u00a0terms<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>11<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22128<\/mn><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Isolate\u00a0the\u00a0second\u00a0radical<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>11<\/mn><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo>(<\/mo><mn>\u22128<\/mn><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>)<\/mo>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Square\u00a0both\u00a0sides<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>22<\/mn><mi>x<\/mi><mo>+<\/mo><mn>121<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>64<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>+<\/mo><msup>\r\n<mrow>\r\n<mo>(<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>)<\/mo>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>16<\/mn><mo>\u2212<\/mo><mn>8<\/mn><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>14<\/mn><mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>8<\/mn><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Combine\u00a0like\u00a0terms<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>11<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22128<\/mn><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Isolate\u00a0the\u00a0second\u00a0radical<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>11<\/mn><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo>(<\/mo><mn>\u22128<\/mn><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>)<\/mo>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Square\u00a0both\u00a0sides<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>22<\/mn><mi>x<\/mi><mo>+<\/mo><mn>121<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>64<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1175462\">Now that both radicals have been eliminated, set the quadratic equal to zero and solve.<\/p>\r\n\r\n<div id=\"fs-id2946619\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>22<\/mn><mi>x<\/mi><mo>+<\/mo><mn>121<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>64<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>128<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>86<\/mn><mi>x<\/mi><mo>+<\/mo><mn>249<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>83<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Factor\u00a0and\u00a0solve<\/mtext><mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>3<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>83<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>22<\/mn><mi>x<\/mi><mo>+<\/mo><mn>121<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>64<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>128<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>86<\/mn><mi>x<\/mi><mo>+<\/mo><mn>249<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>83<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>Factor\u00a0and\u00a0solve<\/mtext><mo>.<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>3<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>83<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1563543\">The proposed solutions are<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>3<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>3<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>83.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>83.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>Check each solution in the original equation.\r\n<div id=\"fs-id2655511\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mo>+<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>4<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mn>9<\/mn>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\r\n<mn>1<\/mn>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mn>3<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>3<\/mn>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mo>+<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>4<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mn>9<\/mn>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\r\n<mn>1<\/mn>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mn>3<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>3<\/mn>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1840544\">One solution is<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>3.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>3.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<p id=\"fs-id1918601\">Check<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>x<\/mi><mo>=<\/mo><mn>83.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>=<\/mo><mn>83.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id1941177\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mo>+<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>4<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>83<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>83<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>169<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\r\n<mrow>\r\n<mn>81<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>13<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>\u2260<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mo>+<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>4<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>83<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>83<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>169<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\r\n<mrow>\r\n<mn>81<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>13<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>\u2260<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2293780\">The only solution is<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>3.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>3.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>We see that\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>83<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>83<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is an extraneous solution.\r\n\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2430499\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#6<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_06_06\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2702970\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2429740\">Solve the equation with two radicals:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msqrt>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>+<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mn>1.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msqrt>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>+<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mn>1.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id3240737\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Solving an Absolute Value Equation<\/h2>\r\n<p id=\"fs-id1335626\">Next, we will learn how to solve an <span id=\"term-00005\" class=\"no-emphasis\" data-type=\"term\">absolute value equation<\/span>. To solve an equation such as<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>|<\/mo> <mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>8<\/mn><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>8<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>we notice that the absolute value will be equal to 8 if the quantity inside the absolute value bars is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>8<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>8<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>or\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>\u22128.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22128.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>This leads to two different equations we can solve independently.\r\n<div id=\"fs-id2638694\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>8<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>or<\/mtext><mspace width=\"2em\"><\/mspace><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22128<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mrow>\r\n<mn>14<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22122<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>7<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22121<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>8<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mrow>\r\n<mspace width=\"2em\"><\/mspace><mtext>or<\/mtext><mspace width=\"2em\"><\/mspace><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22128<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mrow>\r\n<mn>14<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22122<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mn>7<\/mn>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22121<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1280820\">Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.<\/p>\r\n\r\n<div id=\"fs-id2918915\" class=\"ui-has-child-title\" data-type=\"note\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Absolute Value Equations<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1542481\">The absolute value of <em data-effect=\"italics\">x <\/em>is written as<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>|<\/mo> <mi>x<\/mi> <mo>|<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mi>x<\/mi> <mo>|<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>It has the following properties:\r\n<div id=\"fs-id2434805\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mtext>If\u00a0<\/mtext><mi>x<\/mi><mo>\u2265<\/mo><mn>0<\/mn><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mtext>then\u00a0<\/mtext><mo>|<\/mo><mi>x<\/mi><mo>|<\/mo><mo>=<\/mo><mi>x<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mtext>If\u00a0<\/mtext><mi>x<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mtext>then\u00a0<\/mtext><mo>|<\/mo><mi>x<\/mi><mo>|<\/mo><mo>=<\/mo><mi>\u2212<\/mi><mi>x<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mtext>If\u00a0<\/mtext><mi>x<\/mi><mo>\u2265<\/mo><mn>0<\/mn><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mtext>then\u00a0<\/mtext><mo>|<\/mo><mi>x<\/mi><mo>|<\/mo><mo>=<\/mo><mi>x<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mtext>If\u00a0<\/mtext><mi>x<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mtext>then\u00a0<\/mtext><mo>|<\/mo><mi>x<\/mi><mo>|<\/mo><mo>=<\/mo><mi>\u2212<\/mi><mi>x<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2905392\">For real numbers<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>A<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>A<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>B<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>B<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>an equation of the form\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>|<\/mo> <mi>A<\/mi> <mo>|<\/mo><\/mrow><mo>=<\/mo><mi>B<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mi>A<\/mi> <mo>|<\/mo><\/mrow><mo>=<\/mo><mi>B<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>with\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>B<\/mi><mo>\u2265<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>B<\/mi><mo>\u2265<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>will have solutions when\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>A<\/mi><mo>=<\/mo><mi>B<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>A<\/mi><mo>=<\/mo><mi>B<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>or\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>A<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>B<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>A<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>B<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>If\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>B<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>B<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>the equation\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>|<\/mo> <mi>A<\/mi> <mo>|<\/mo><\/mrow><mo>=<\/mo><mi>B<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mi>A<\/mi> <mo>|<\/mo><\/mrow><mo>=<\/mo><mi>B<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>has no solution.\r\n<p id=\"fs-id1408140\">An <span id=\"term-00006\" data-type=\"term\">absolute value equation<\/span> in the form<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>|<\/mo> <mrow>\r\n<mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mi>c<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mrow>\r\n<mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mi>c<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>has the following properties:\r\n<div id=\"fs-id2655133\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mtext>If\u00a0\u00a0<\/mtext><mi>c<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo>,<\/mo><mo>|<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>|<\/mo><mo>=<\/mo><mi>c<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>has\u00a0no\u00a0solution<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mtext>If\u00a0\u00a0<\/mtext><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo><mo>|<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>|<\/mo><mo>=<\/mo><mi>c<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>has\u00a0one\u00a0solution<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mtext>If\u00a0\u00a0<\/mtext><mi>c<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo><mo>|<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>|<\/mo><mo>=<\/mo><mi>c<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>has\u00a0two\u00a0solutions<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mtext>If\u00a0\u00a0<\/mtext><mi>c<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo>,<\/mo><mo>|<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>|<\/mo><mo>=<\/mo><mi>c<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>has\u00a0no\u00a0solution<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mtext>If\u00a0\u00a0<\/mtext><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo><mo>|<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>|<\/mo><mo>=<\/mo><mi>c<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>has\u00a0one\u00a0solution<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mtext>If\u00a0\u00a0<\/mtext><mi>c<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo><mo>|<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>|<\/mo><mo>=<\/mo><mi>c<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>has\u00a0two\u00a0solutions<\/mtext><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3038796\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1793504\"><strong>Given an absolute value equation, solve it.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id2938958\" type=\"1\">\r\n \t<li>Isolate the absolute value expression on one side of the equal sign.<\/li>\r\n \t<li>If\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>c<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>c<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>write and solve two equations:\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>=<\/mo><mi>c<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>=<\/mo><mi>c<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>c<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>c<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_06_08\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">8<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id3105697\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1836188\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving Absolute Value Equations<\/h3>\r\n<p id=\"fs-id1836194\">Solve the following absolute value equations:<\/p>\r\n\r\n<ol id=\"fs-id2370595\" class=\"circled\" type=\"1\">\r\n \t<li><span class=\"token\">\u24d0<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>8<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>8<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li><span class=\"token\">\u24d1<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>\u22129<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>\u22129<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li><span class=\"token\">\u24d2<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>4<\/mn><mo>=<\/mo><mn>6<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>4<\/mn><mo>=<\/mo><mn>6<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li><span class=\"token\">\u24d3<\/span>\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>\u22125<\/mn><mi>x<\/mi><mo>+<\/mo><mn>10<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>\u22125<\/mn><mi>x<\/mi><mo>+<\/mo><mn>10<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id2683627\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<ol id=\"fs-id2683629\" class=\"circled\" type=\"1\">\r\n \t<li><span class=\"token\">\u24d0<\/span>\r\n<p id=\"fs-id1832720\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>8<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>8<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<p id=\"fs-id2389886\">Write two equations and solve each:<\/p>\r\n\r\n<div id=\"eip-id1170402604810\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow><mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>8<\/mn>\r\n<\/mtd><\/mtr><\/mtable>\r\n\r\n&nbsp;\r\n\r\n<mtd columnalign=\"right\">\r\n<mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>\u22128<\/mn>\r\n<\/mtd>\r\n\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mn>6<\/mn><mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>4<\/mn>\r\n<\/mtd><\/mtr>\r\n\r\n&nbsp;\r\n\r\n<mtd columnalign=\"right\">\r\n<mn>6<\/mn><mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>\u221212<\/mn>\r\n<\/mtd>\r\n\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mtd><\/mtr>\r\n\r\n&nbsp;\r\n\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>\u22122<\/mn>\r\n<\/mtd>\r\n\r\n\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>8<\/mn>\r\n<\/mtd>&nbsp;\r\n\r\n<mtd columnalign=\"right\">\r\n<mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>\u22128<\/mn>\r\n<\/mtd>\r\n\r\n<\/mtr><mtr>\r\n<mtd columnalign=\"right\">\r\n<mn>6<\/mn><mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>4<\/mn>\r\n<\/mtd><\/mtr>\r\n\r\n&nbsp;\r\n\r\n<mtd columnalign=\"right\">\r\n<mn>6<\/mn><mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>\u221212<\/mn>\r\n<\/mtd>\r\n\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mtd><\/mtr>\r\n\r\n&nbsp;\r\n\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>\u22122<\/mn>\r\n<\/mtd>\r\n\r\n<\/mtable><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id3203361\">The two solutions are<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22122.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22122.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/li>\r\n \t<li><span class=\"token\">\u24d1<\/span>\r\n<p id=\"fs-id1879343\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>\u22129<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>\u22129<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<p id=\"fs-id2293707\">There is no solution as an absolute value cannot be negative.<\/p>\r\n<\/li>\r\n \t<li><span class=\"token\">\u24d2<\/span>\r\n<p id=\"fs-id2888792\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>4<\/mn><mo>=<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>4<\/mn><mo>=<\/mo><mn>6<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<p id=\"fs-id2436076\">Isolate the absolute value expression and then write two equations.<\/p>\r\n\r\n<div id=\"fs-id3274908\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>6<\/mn><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow> <mo>|<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>10<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>10<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>\u221210<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>15<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22125<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>5<\/mn><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>6<\/mn><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow> <mo>|<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>10<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>10<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>\u221210<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>15<\/mn><\/mrow><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22125<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>5<\/mn><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\"><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\r\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2498828\">There are two solutions:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>5<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>5<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>5<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/li>\r\n \t<li><span class=\"token\">\u24d3<\/span>\r\n<p id=\"fs-id1340685\">(d)<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>\u22125<\/mn><mi>x<\/mi><mo>+<\/mo><mn>10<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>\u22125<\/mn><mi>x<\/mi><mo>+<\/mo><mn>10<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<p id=\"fs-id2016618\">The equation is set equal to zero, so we have to write only one equation.<\/p>\r\n\r\n<div id=\"fs-id2431537\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>\u22125<\/mn><mi>x<\/mi><mo>+<\/mo><mn>10<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mn>0<\/mn><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mn>\u22125<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mrow><mn>\u221210<\/mn><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mi>x<\/mi><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mn>2<\/mn><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>\u22125<\/mn><mi>x<\/mi><mo>+<\/mo><mn>10<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mn>0<\/mn><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mn>\u22125<\/mn><mi>x<\/mi><\/mrow><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mrow><mn>\u221210<\/mn><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mi>x<\/mi><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mn>2<\/mn><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1351555\">There is one solution:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>2.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>2.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1535632\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#7<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_06_07\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2515512\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2515513\">Solve the absolute value equation:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mo>|<\/mo>\r\n<mn>1<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi>\r\n<mo>|<\/mo><mo>+<\/mo><mn>8<\/mn><mo>=<\/mo><mn>13.<\/mn>&nbsp;\r\n\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>|<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><mo>|<\/mo><mo>+<\/mo><mn>8<\/mn><mo>=<\/mo><mn>13.<\/mn><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2653809\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Solving Other Types of Equations<\/h2>\r\n<p id=\"fs-id3169548\">There are many other types of equations in addition to the ones we have discussed so far. We will see more of them throughout the text. Here, we will discuss equations that are in quadratic form, and rational equations that result in a quadratic.<\/p>\r\n\r\n<section id=\"fs-id1923869\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Solving Equations in Quadratic Form<\/h3>\r\n<p id=\"fs-id1923873\"><strong>Equations in quadratic form <\/strong>are equations with three terms. The first term has a power other than 2. The middle term has an exponent that is one-half the exponent of the leading term. The third term is a constant. We can solve equations in this form as if they were quadratic. A few examples of these equations include<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msup>\r\n<mi>x<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>5<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>6<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>7<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>8<\/mn><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\r\n<mi>x<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>5<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>6<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>7<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>8<\/mn><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>In each one, doubling the exponent of the middle term equals the exponent on the leading term. We can solve these equations by substituting a variable for the middle term.\r\n<div id=\"fs-id1834422\" class=\"ui-has-child-title\" data-type=\"note\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Quadratic Form<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id3094567\">If the exponent on the middle term is one-half of the exponent on the leading term, we have an <span id=\"term-00007\" data-type=\"term\">equation in quadratic form<\/span>, which we can solve as if it were a quadratic. We substitute a variable for the middle term to solve equations in quadratic form.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1954745\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1385726\"><strong>Given an equation quadratic in form, solve it.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1385730\" type=\"1\">\r\n \t<li>Identify the exponent on the leading term and determine whether it is double the exponent on the middle term.<\/li>\r\n \t<li>If it is, substitute a variable, such as <em data-effect=\"italics\">u<\/em>, for the variable portion of the middle term.<\/li>\r\n \t<li>Rewrite the equation so that it takes on the standard form of a quadratic.<\/li>\r\n \t<li>Solve using one of the usual methods for solving a quadratic.<\/li>\r\n \t<li>Replace the substitution variable with the original term.<\/li>\r\n \t<li>Solve the remaining equation.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_06_09\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">9<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1357813\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1357816\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving a Fourth-degree Equation in Quadratic Form<\/h3>\r\n<p id=\"fs-id2486573\">Solve this fourth-degree equation:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>2<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>2<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1229226\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id1229229\">This equation fits the main criteria, that the power on the leading term is double the power on the middle term. Next, we will make a substitution for the variable term in the middle. Let<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>u<\/mi><mo>=<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>u<\/mi><mo>=<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>Rewrite the equation in <em data-effect=\"italics\">u<\/em>.\r\n<div id=\"fs-id3155491\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><msup>\r\n<mi>u<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>2<\/mn><mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><msup>\r\n<mi>u<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>2<\/mn><mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2371956\">Now solve the quadratic.<\/p>\r\n\r\n<div id=\"fs-id2371960\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><msup>\r\n<mi>u<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>2<\/mn><mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mn>0<\/mn><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mi>u<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><msup>\r\n<mi>u<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>2<\/mn><mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mn>0<\/mn><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>3<\/mn><mi>u<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2308574\">Solve each factor and replace the original term for <em data-effect=\"italics\">u.<\/em><\/p>\r\n\r\n<div id=\"fs-id2797268\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><mi>u<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><mi>u<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22121<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>u<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u00b1<\/mo><mi>i<\/mi><msqrt>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><mi>u<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>3<\/mn><mi>u<\/mi><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22121<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>u<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u00b1<\/mo><mi>i<\/mi><msqrt>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<div id=\"fs-id1335933\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>u<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>1<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>1<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u00b11<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>u<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>1<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>1<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u00b11<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2980376\">The solutions are<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mo>\u00b1<\/mo><mi>i<\/mi><msqrt>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mo>\u00b1<\/mo><mi>i<\/mi><msqrt>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mo>\u00b1<\/mo><mn>1.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mo>\u00b1<\/mo><mn>1.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1929412\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#8<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_06_08\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2514218\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2514219\">Solve using substitution:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msup>\r\n<mi>x<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>8<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>9<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\r\n<mi>x<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>8<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>9<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_06_10\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">10<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id2444359\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2444361\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving an Equation in Quadratic Form Containing a Binomial<\/h3>\r\n<p id=\"fs-id2029548\">Solve the equation in quadratic form:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>11<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mn>12<\/mn><mo>=<\/mo><mn>0.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>11<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mn>12<\/mn><mo>=<\/mo><mn>0.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id2382010\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id2382012\">This equation contains a binomial in place of the single variable. The tendency is to expand what is presented. However, recognizing that it fits the criteria for being in quadratic form makes all the difference in the solving process. First, make a substitution, letting<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>u<\/mi><mo>=<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>u<\/mi><mo>=<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>Then rewrite the equation in <em data-effect=\"italics\">u.<\/em>\r\n<div id=\"fs-id1929605\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>u<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>11<\/mn><mi>u<\/mi><mo>\u2212<\/mo><mn>12<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>u<\/mi><mo>+<\/mo><mn>12<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>u<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>11<\/mn><mi>u<\/mi><mo>\u2212<\/mo><mn>12<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>u<\/mi><mo>+<\/mo><mn>12<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1691070\">Solve using the zero-factor property and then replace <em data-effect=\"italics\">u<\/em> with the original expression.<\/p>\r\n\r\n<div id=\"fs-id1516932\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>u<\/mi><mo>+<\/mo><mn>12<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>u<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u221212<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u221212<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u221214<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>u<\/mi><mo>+<\/mo><mn>12<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>u<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u221212<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u221212<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u221214<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1720759\">The second factor results in<\/p>\r\n\r\n<div id=\"fs-id2485990\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>u<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>1<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>1<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22121<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>u<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>1<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>1<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22121<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1943470\">We have two solutions:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mn>\u221214<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u221214<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22121.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22121.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1736480\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#9<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_06_09\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1952007\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1952008\">Solve:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mn>21<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mn>21<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2893053\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Solving Rational Equations Resulting in a Quadratic<\/h3>\r\n<p id=\"fs-id1679917\">Earlier, we solved rational equations. Sometimes, solving a rational equation results in a quadratic. When this happens, we continue the solution by simplifying the quadratic equation by one of the methods we have seen. It may turn out that there is no solution.<\/p>\r\n\r\n<div id=\"Example_02_06_11\" class=\"ui-has-child-title\" data-type=\"example\"><header>\r\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span>\r\n<span class=\"os-number\">11<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id1970290\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2508561\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving a Rational Equation Leading to a Quadratic<\/h3>\r\n<p id=\"fs-id2508566\">Solve the following rational equation:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>8<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>8<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id2797285\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\"><summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary><section class=\"ui-body\" role=\"alert\">\r\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\r\n<div class=\"os-solution-container\">\r\n<p id=\"fs-id2797287\">We want all denominators in factored form to find the LCD. Two of the denominators cannot be factored further. However,<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mn>\u22121<\/mn><mo>=<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mn>\u22121<\/mn><mo>=<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>Then, the LCD is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>Next, we multiply the whole equation by the LCD.\r\n<div id=\"fs-id2441856\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mrow><mo>[<\/mo> <mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>\u22124<\/mn><mi>x<\/mi><\/mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mfrac>\r\n<\/mrow> <mo>]<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>[<\/mo> <mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>\u22128<\/mn><\/mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mfrac>\r\n<\/mrow> <mo>]<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>\u22124<\/mn><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22128<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>\u22124<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22128<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>\u22124<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>\u22124<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>\u22124<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22121<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>1<\/mn>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtable>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mrow><mo>[<\/mo> <mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>\u22124<\/mn><mi>x<\/mi><\/mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>4<\/mn>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mfrac>\r\n<\/mrow> <mo>]<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mrow><mo>[<\/mo> <mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>\u22128<\/mn><\/mrow>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mfrac>\r\n<\/mrow> <mo>]<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>\u22124<\/mn><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22128<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>\u22124<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22128<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>\u22124<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>\u22124<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>\u22124<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>0<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>\u22121<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd>\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mn>1<\/mn>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2980350\">In this case, either solution produces a zero in the denominator in the original equation. Thus, there is no solution.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2906702\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\"><header>\r\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span>\r\n<span class=\"os-number\">#10<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_02_06_10\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1751154\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1751155\">Solve<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mi>x<\/mi>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>+<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mi>x<\/mi>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2665007\" class=\"media-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Media\"><header>\r\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"Media\"><span class=\"os-title-label\">Media<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1822998\">Access these online resources for additional instruction and practice with different types of equations.<\/p>\r\n\r\n<ul id=\"fs-id1823002\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/rateqnosoln\" target=\"_blank\" rel=\"noopener nofollow\">Rational Equation with no Solution<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/ratexprecpexp\" target=\"_blank\" rel=\"noopener nofollow\">Solving equations with rational exponents using reciprocal powers<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/radeqsolvepart1\" target=\"_blank\" rel=\"noopener nofollow\">Solving radical equations part 1 of 2<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/radeqsolvepart2\" target=\"_blank\" rel=\"noopener nofollow\">Solving radical equations part 2 of 2<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section>\r\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\r\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">2.6 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id3146179\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id3146185\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id2028956\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2028957\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2028956-solution\">1<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2028958\">In a radical equation, what does it mean if a number is an extraneous solution?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3256935\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3256936\" data-type=\"problem\"><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3256937\">Explain why possible solutions <em data-effect=\"italics\">must<\/em> be checked in radical equations.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2501234\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2501235\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2501234-solution\">3<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2501236\">Your friend tries to calculate the value<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mo>\u2212<\/mo><msup>\r\n<mn>9<\/mn>\r\n<mrow>\r\n<mstyle scriptlevel=\"+1\">\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mstyle>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mo>\u2212<\/mo><msup>\r\n<mn>9<\/mn>\r\n<mrow>\r\n<mstyle scriptlevel=\"+1\">\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mstyle>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and keeps getting an ERROR message. What mistake are they probably making?\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1941430\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1941431\" data-type=\"problem\"><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1941432\">Explain why<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mrow><mo>|<\/mo> <mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>\u22127<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>\u22127<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>has no solutions.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1354283\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1354284\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1354283-solution\">5<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1354285\">Explain how to change a rational exponent into the correct radical expression.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id3018948\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<p id=\"fs-id2628823\">For the following exercises, solve the rational exponent equation. Use factoring where necessary.<\/p>\r\n\r\n<div id=\"fs-id2628826\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2628827\" data-type=\"problem\"><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2628828\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><mn>16<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><mn>16<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1973815\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1973816\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1973815-solution\">7<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1973817\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><mn>27<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><mn>27<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2432476\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2432477\" data-type=\"problem\"><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2432478\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3040226\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3040227\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3040226-solution\">9<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1296521\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><mn>8<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><mn>8<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3060328\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3060329\" data-type=\"problem\"><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3060330\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><mn>4<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><mn>4<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1763592\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1763593\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1763592-solution\">11<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1763594\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>5<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>+<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>5<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>+<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1709842\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1709843\" data-type=\"problem\"><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1709844\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>4<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>7<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>4<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id2382870\">For the following exercises, solve the following polynomial equations by grouping and factoring.<\/p>\r\n\r\n<div id=\"fs-id2382874\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2382875\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2382874-solution\">13<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2382876\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>2<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>2<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3111192\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3111193\" data-type=\"problem\"><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3111194\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>6<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>27<\/mn><mi>x<\/mi><mo>+<\/mo><mn>54<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>6<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>27<\/mn><mi>x<\/mi><mo>+<\/mo><mn>54<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1393262\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1393263\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1393262-solution\">15<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1393264\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>4<\/mn><msup>\r\n<mi>y<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>9<\/mn><mi>y<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>4<\/mn><msup>\r\n<mi>y<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>9<\/mn><mi>y<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2529330\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2529331\" data-type=\"problem\"><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2529332\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>25<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>75<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>25<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>75<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1582399\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1582400\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1582399-solution\">17<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1582402\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>m<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><msup>\r\n<mi>m<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mi>m<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>m<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><msup>\r\n<mi>m<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mi>m<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id586482\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id586483\" data-type=\"problem\"><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2998491\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>5<\/mn>\r\n<\/msup>\r\n<mn>\u221214<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>5<\/mn>\r\n<\/msup>\r\n<mn>\u221214<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2643100\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2643101\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2643100-solution\">19<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2643102\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>5<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>45<\/mn><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>18<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>5<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>3<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>45<\/mn><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>18<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id2917413\">For the following exercises, solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions.<\/p>\r\n\r\n<div id=\"fs-id2917418\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2294086\" data-type=\"problem\"><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2294087\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mo>\u2212<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mo>\u2212<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2332670\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2332671\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2332670-solution\">21<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2332672\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mn>5<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mn>5<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2375964\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2375966\" data-type=\"problem\"><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2375967\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2486495\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2486496\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2486495-solution\">23<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2486497\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>3<\/mn><mi>t<\/mi><mo>+<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mn>7<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>3<\/mn><mi>t<\/mi><mo>+<\/mo><mn>5<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mn>7<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1551969\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1551970\" data-type=\"problem\"><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1551971\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mi>t<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>+<\/mo><mn>9<\/mn><mo>=<\/mo><mn>7<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msqrt>\r\n<mrow>\r\n<mi>t<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>+<\/mo><mn>9<\/mn><mo>=<\/mo><mn>7<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3256844\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3256845\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3256844-solution\">25<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3256846\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>12<\/mn><mo>\u2212<\/mo><mi>x<\/mi><\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mi>x<\/mi><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>12<\/mn><mo>\u2212<\/mo><mi>x<\/mi><\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mi>x<\/mi><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1357710\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1357711\" data-type=\"problem\"><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1357712\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>\u2212<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>\u2212<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mn>2<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1441558\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1441559\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1441558-solution\">27<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1441560\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>+<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>+<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mn>1<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1951698\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1951699\" data-type=\"problem\"><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1951700\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mo>\u2212<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mo>\u2212<\/mo><msqrt>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mn>1<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1786258\">For the following exercises, solve the equation involving absolute value.<\/p>\r\n\r\n<div id=\"fs-id1150948\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1150949\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1150948-solution\">29<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1150950\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>8<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>8<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2307448\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2307449\" data-type=\"problem\"><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2307450\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>\u22122<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>\u22122<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2410873\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2410874\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2410873-solution\">31<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2410875\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>1<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>5<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>1<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>5<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2428664\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2428665\" data-type=\"problem\"><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2428666\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>3<\/mn><mo>=<\/mo><mn>6<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>3<\/mn><mo>=<\/mo><mn>6<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1717824\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1717825\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1717824-solution\">33<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1717826\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>7<\/mn><mo>=<\/mo><mn>\u22122<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>7<\/mn><mo>=<\/mo><mn>\u22122<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3165049\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3165050\" data-type=\"problem\"><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3165051\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>2<\/mn><mo>=<\/mo><mn>\u22123<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>2<\/mn><mo>=<\/mo><mn>\u22123<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2382990\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2382991\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2382990-solution\">35<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2382992\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2643356\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2643357\" data-type=\"problem\"><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2643358\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo>\u2212<\/mo><mrow><mo>|<\/mo> <mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>\u22123<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo>\u2212<\/mo><mrow><mo>|<\/mo> <mrow>\r\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>\u22123<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id2665057\">For the following exercises, solve the equation by identifying the quadratic form. Use a substitute variable and find all real solutions by factoring.<\/p>\r\n\r\n<div id=\"fs-id2665062\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2665063\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2665062-solution\">37<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2386875\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>10<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>9<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>4<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>10<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>9<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1715518\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1715519\" data-type=\"problem\"><span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1715520\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>4<\/mn><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>t<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>9<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>t<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>\u22122<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>4<\/mn><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>t<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>9<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>t<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>\u22122<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1577029\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2029121\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1577029-solution\">39<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2029122\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mn>12<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mn>12<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1418519\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1418520\" data-type=\"problem\"><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1418521\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>8<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mn>9<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>8<\/mn><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mn>9<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2797000\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2797001\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2797000-solution\">41<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2797002\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2390141\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Extensions<\/h3>\r\n<p id=\"fs-id2390146\">For the following exercises, solve for the unknown variable.<\/p>\r\n\r\n<div id=\"fs-id2390150\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2390151\" data-type=\"problem\"><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2390152\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mn>\u22122<\/mn><\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mn>\u22121<\/mn><\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>12<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mn>\u22122<\/mn><\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mrow>\r\n<mn>\u22121<\/mn><\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>12<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1726737\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1726738\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1726737-solution\">43<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1726739\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mi>x<\/mi> <mo>|<\/mo><\/mrow>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msqrt>\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mi>x<\/mi> <mo>|<\/mo><\/mrow>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>=<\/mo><mi>x<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2412235\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2412236\" data-type=\"problem\"><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2412237\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>t<\/mi>\r\n<mrow>\r\n<mn>10<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>2<\/mn><msup>\r\n<mi>t<\/mi>\r\n<mn>5<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>t<\/mi>\r\n<mrow>\r\n<mn>10<\/mn>\r\n<\/mrow>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>2<\/mn><msup>\r\n<mi>t<\/mi>\r\n<mn>5<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1276816\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1276817\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1276816-solution\">45<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1276818\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>36<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>12<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>|<\/mo> <mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>36<\/mn>\r\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>12<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1976510\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Real-World Applications<\/h3>\r\n<p id=\"fs-id1976516\">For the following exercises, use the model for the period of a pendulum,<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>T<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>T<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>such that\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>T<\/mi><mo>=<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><msqrt>\r\n<mrow>\r\n<mfrac>\r\n<mi>L<\/mi>\r\n<mi>g<\/mi>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>T<\/mi><mo>=<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><msqrt>\r\n<mrow>\r\n<mfrac>\r\n<mi>L<\/mi>\r\n<mi>g<\/mi>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>where the length of the pendulum is <em data-effect=\"italics\">L<\/em> and the acceleration due to gravity is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>g<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>g<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id2500088\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2500089\" data-type=\"problem\"><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2500090\">If the acceleration due to gravity is 9.8 m\/s<sup>2<\/sup> and the period equals 1 s, find the length to the nearest cm (100 cm = 1 m).<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2947584\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2947585\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2947584-solution\">47<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1762772\">If the gravity is 32 ft\/s<sup>2<\/sup> and the period equals 1 s, find the length to the nearest in. (12 in. = 1 ft). Round your answer to the nearest in.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1591913\">For the following exercises, use a model for body surface area, BSA, such that<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>B<\/mi><mi>S<\/mi><mi>A<\/mi><mo>=<\/mo><msqrt>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mi>w<\/mi><mi>h<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mn>3600<\/mn>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>B<\/mi><mi>S<\/mi><mi>A<\/mi><mo>=<\/mo><msqrt>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mi>w<\/mi><mi>h<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mn>3600<\/mn>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/msqrt>\r\n<mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>where <em data-effect=\"italics\">w<\/em> = weight in kg and <em data-effect=\"italics\">h<\/em> = height in cm.\r\n<div id=\"fs-id3070467\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3070468\" data-type=\"problem\"><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3070470\">Find the height of a 72-kg female to the nearest cm whose<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>B<\/mi><mi>S<\/mi><mi>A<\/mi><mo>=<\/mo><mn>1.8.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>B<\/mi><mi>S<\/mi><mi>A<\/mi><mo>=<\/mo><mn>1.8.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1591172\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1591174\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1591172-solution\">49<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1591175\">Find the weight of a 177-cm male to the nearest kg whose<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>B<\/mi><mi>S<\/mi><mi>A<\/mi><mo>=<\/mo><mn>2.1.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>B<\/mi><mi>S<\/mi><mi>A<\/mi><mo>=<\/mo><mn>2.1.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>\r\n<\/div>\r\n<\/div>","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_b88d40b3-a0d5-4a8f-8b2e-85993b911cc0\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\n<div class=\"ui-has-child-title\" data-type=\"abstract\">\n<header>\n<h2 data-type=\"title\">Learning Objectives<\/h2>\n<\/header>\n<section>\n<p id=\"para-00001\">In this section, you will:<\/p>\n<ul id=\"list-00001\">\n<li>Solve equations involving rational exponents.<\/li>\n<li>Solve equations using factoring.<\/li>\n<li>Solve radical equations.<\/li>\n<li>Solve absolute value equations.<\/li>\n<li>Solve other types of equations.<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<p id=\"fs-id1519532\">We have solved linear equations, rational equations, and quadratic equations using several methods. However, there are many other types of equations, and we will investigate a few more types in this section. We will look at equations involving rational exponents, polynomial equations, radical equations, absolute value equations, equations in quadratic form, and some rational equations that can be transformed into quadratics. Solving any equation, however, employs the same basic algebraic rules. We will learn some new techniques as they apply to certain equations, but the algebra never changes.<\/p>\n<section id=\"fs-id2485804\" data-depth=\"1\">\n<h2 data-type=\"title\">Solving Equations Involving Rational Exponents<\/h2>\n<p id=\"fs-id1274976\">Rational exponents are exponents that are fractions, where the numerator is a power and the denominator is a root. For example,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msup>\n<mrow>\n<mn>16<\/mn>\n<\/mrow>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\n<mrow>\n<mn>16<\/mn>\n<\/mrow>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>is another way of writing<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msqrt>\n<mn>16<\/mn>\n<\/msqrt>\n<mo>;<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msqrt>\n<mn>16<\/mn>\n<\/msqrt>\n<mo>;<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mn>8<\/mn>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mn>8<\/mn>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>is another way of writing<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mtext>\u200b<\/mtext><mspace width=\"0.5em\"><\/mspace><mroot>\n<mn>8<\/mn>\n<mn>3<\/mn>\n<\/mroot>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtext>\u200b<\/mtext><mspace width=\"0.5em\"><\/mspace><mroot>\n<mn>8<\/mn>\n<mn>3<\/mn>\n<\/mroot>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>The ability to work with rational exponents is a useful skill, as it is highly applicable in calculus.<\/p>\n<p id=\"fs-id1723049\">We can solve equations in which a variable is raised to a rational exponent by raising both sides of the equation to the reciprocal of the exponent. The reason we raise the equation to the reciprocal of the exponent is because we want to eliminate the exponent on the variable term, and a number multiplied by its reciprocal equals 1. For example,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and so on.<\/p>\n<div id=\"fs-id1319605\" class=\"ui-has-child-title\" data-type=\"note\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Rational Exponents<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1532274\">A rational exponent indicates a power in the numerator and a root in the denominator. There are multiple ways of writing an expression, a variable, or a number with a rational exponent:<\/p>\n<div id=\"fs-id1512276\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mrow>\n<mfrac>\n<mi>m<\/mi>\n<mi>n<\/mi>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mi>n<\/mi>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mi>m<\/mi>\n<\/msup>\n<mo>=<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mi>m<\/mi>\n<\/msup>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mi>n<\/mi>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><mroot>\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mi>m<\/mi>\n<\/msup>\n<\/mrow>\n<mi>n<\/mi>\n<\/mroot>\n<mo>=<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mroot>\n<mi>a<\/mi>\n<mi>n<\/mi>\n<\/mroot>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mi>m<\/mi>\n<\/msup>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>a<\/mi>\n<mrow>\n<mfrac>\n<mi>m<\/mi>\n<mi>n<\/mi>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mi>n<\/mi>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mi>m<\/mi>\n<\/msup>\n<mo>=<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mi>m<\/mi>\n<\/msup>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mi>n<\/mi>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><mroot>\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mi>m<\/mi>\n<\/msup>\n<\/mrow>\n<mi>n<\/mi>\n<\/mroot>\n<mo>=<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mroot>\n<mi>a<\/mi>\n<mi>n<\/mi>\n<\/mroot>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mi>m<\/mi>\n<\/msup>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_06_01\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">1<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1458449\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2592723\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Evaluating a Number Raised to a Rational Exponent<\/h3>\n<p id=\"fs-id1239738\">Evaluate<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msup>\n<mn>8<\/mn>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\n<mn>8<\/mn>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id2522284\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id2422284\">Whether we take the root first or the power first depends on the number. It is easy to find the cube root of 8, so rewrite<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msup>\n<mn>8<\/mn>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\n<mn>8<\/mn>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mn>8<\/mn>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mn>8<\/mn>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id1227998\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mn>8<\/mn>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>4<\/mn><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mn>8<\/mn>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>4<\/mn><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1514351\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#1<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_06_01\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1336704\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id2400251\">Evaluate<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msup>\n<mrow>\n<mn>64<\/mn>\n<\/mrow>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\n<mrow>\n<mn>64<\/mn>\n<\/mrow>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_06_02\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">2<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id2385021\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3264211\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solve the Equation Including a Variable Raised to a Rational Exponent<\/h3>\n<p id=\"fs-id1198214\">Solve the equation in which a variable is raised to a rational exponent:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>5<\/mn>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><mn>32.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>5<\/mn>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><mn>32.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id2431600\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1569151\">The way to remove the exponent on <em data-effect=\"italics\">x<\/em> is by raising both sides of the equation to a power that is the reciprocal of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mfrac>\n<mn>5<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mfrac>\n<mn>5<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>which is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mfrac>\n<mn>4<\/mn>\n<mn>5<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mfrac>\n<mn>4<\/mn>\n<mn>5<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id1277511\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>5<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>32<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>5<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mrow>\n<mfrac>\n<mn>4<\/mn>\n<mn>5<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>32<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mrow>\n<mfrac>\n<mn>4<\/mn>\n<mn>5<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mn>4<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>The\u00a0fifth\u00a0root\u00a0of\u00a032\u00a0is\u00a02.<\/mtext><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>16<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>5<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>32<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>5<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mrow>\n<mfrac>\n<mn>4<\/mn>\n<mn>5<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>32<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mrow>\n<mfrac>\n<mn>4<\/mn>\n<mn>5<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mn>4<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>The\u00a0fifth\u00a0root\u00a0of\u00a032\u00a0is\u00a02.<\/mtext><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>16<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1759744\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#2<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_06_02\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2293541\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1314798\">Solve the equation<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><mn>125.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><mn>125.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_06_03\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">3<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id2437878\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1335480\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving an Equation Involving Rational Exponents and Factoring<\/h3>\n<p id=\"fs-id1269017\">Solve<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1522050\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1296613\">This equation involves rational exponents as well as factoring rational exponents. Let us take this one step at a time. First, put the variable terms on one side of the equal sign and set the equation equal to zero.<\/p>\n<div id=\"fs-id1400272\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>0<\/mn><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>0<\/mn><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1368636\">Now, it looks like we should factor the left side, but what do we factor out? We can always factor the term with the lowest exponent. Rewrite<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>Then, factor out<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>from both terms on the left.<\/p>\n<div id=\"fs-id1441452\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1933337\">Where did<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>come from? Remember, when we multiply two numbers with the same base, we add the exponents. Therefore, if we multiply<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>back in using the distributive property, we get the expression we had before the factoring, which is what should happen. We need an exponent such that when added to<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mfrac>\n<mn>2<\/mn>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mfrac>\n<mn>2<\/mn>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>equals<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>Thus, the exponent on <em data-effect=\"italics\">x <\/em>in the parentheses is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p id=\"fs-id1891109\">Let us continue. Now we have two factors and can use the zero factor theorem.<\/p>\n<div id=\"fs-id1422353\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>1<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Divide\u00a0both\u00a0sides\u00a0by\u00a03<\/mtext><mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>4<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>4<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Raise\u00a0both\u00a0sides\u00a0to\u00a0the\u00a0reciprocal\u00a0of\u00a0<\/mtext><mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mrow>\n<mn>81<\/mn><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>1<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Divide\u00a0both\u00a0sides\u00a0by\u00a03<\/mtext><mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>4<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>4<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Raise\u00a0both\u00a0sides\u00a0to\u00a0the\u00a0reciprocal\u00a0of\u00a0<\/mtext><mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mrow>\n<mn>81<\/mn><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2957928\">The two solutions are<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>0<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mrow>\n<mn>81<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mn>1<\/mn>\n<mrow>\n<mn>81<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1267637\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#3<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_06_03\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1411367\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1441991\">Solve:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><mn>8.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><mn>8.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2945385\" data-depth=\"1\">\n<h2 data-type=\"title\">Solving Equations Using Factoring<\/h2>\n<p id=\"fs-id2521696\">We have used factoring to solve quadratic equations, but it is a technique that we can use with many types of polynomial equations, which are equations that contain a string of terms including numerical coefficients and variables. When we are faced with an equation containing polynomials of degree higher than 2, we can often solve them by factoring.<\/p>\n<div id=\"fs-id1786047\" class=\"ui-has-child-title\" data-type=\"note\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Polynomial Equations<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id3231810\">A polynomial of degree <em data-effect=\"italics\">n <\/em>is an expression of the type<\/p>\n<div id=\"fs-id2905918\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<msub>\n<mi>a<\/mi>\n<mi>n<\/mi>\n<\/msub>\n<msup>\n<mi>x<\/mi>\n<mi>n<\/mi>\n<\/msup>\n<mo>+<\/mo><msub>\n<mi>a<\/mi>\n<mrow>\n<mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/msub>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/msup>\n<mo>+<\/mo><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>\u22c5<\/mo><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>\u22c5<\/mo><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>\u22c5<\/mo><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>+<\/mo><msub>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msub>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><msub>\n<mi>a<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<mi>x<\/mi><mo>+<\/mo><msub>\n<mi>a<\/mi>\n<mn>0<\/mn>\n<\/msub>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msub>\n<mi>a<\/mi>\n<mi>n<\/mi>\n<\/msub>\n<msup>\n<mi>x<\/mi>\n<mi>n<\/mi>\n<\/msup>\n<mo>+<\/mo><msub>\n<mi>a<\/mi>\n<mrow>\n<mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/msub>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mi>n<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/msup>\n<mo>+<\/mo><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>\u22c5<\/mo><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>\u22c5<\/mo><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>\u22c5<\/mo><mspace width=\"0.5em\"><\/mspace><mspace width=\"0.5em\"><\/mspace><mo>+<\/mo><msub>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msub>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><msub>\n<mi>a<\/mi>\n<mn>1<\/mn>\n<\/msub>\n<mi>x<\/mi><mo>+<\/mo><msub>\n<mi>a<\/mi>\n<mn>0<\/mn>\n<\/msub>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2444064\">where <em data-effect=\"italics\">n<\/em> is a positive integer and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msub>\n<mi>a<\/mi>\n<mi>n<\/mi>\n<\/msub>\n<mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub>\n<mi>a<\/mi>\n<mn>0<\/mn>\n<\/msub>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msub>\n<mi>a<\/mi>\n<mi>n<\/mi>\n<\/msub>\n<mo>,<\/mo><mo>\u2026<\/mo><mo>,<\/mo><msub>\n<mi>a<\/mi>\n<mn>0<\/mn>\n<\/msub>\n<\/mrow><\/annotation-xml><\/semantics><\/math>are real numbers and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msub>\n<mi>a<\/mi>\n<mi>n<\/mi>\n<\/msub>\n<mo>\u2260<\/mo><mn>0.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msub>\n<mi>a<\/mi>\n<mi>n<\/mi>\n<\/msub>\n<mo>\u2260<\/mo><mn>0.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p id=\"fs-id2931289\">Setting the polynomial equal to zero gives a <span id=\"term-00001\" data-type=\"term\">polynomial equation<\/span>. The total number of solutions (real and complex) to a polynomial equation is equal to the highest exponent <em data-effect=\"italics\">n<\/em>.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_06_04\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">4<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1996696\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2501149\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving a Polynomial by Factoring<\/h3>\n<p id=\"fs-id1391817\">Solve the polynomial by factoring:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>5<\/mn><msup>\n<mi>x<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>80<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>5<\/mn><msup>\n<mi>x<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>80<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1517483\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1419163\">First, set the equation equal to zero. Then factor out what is common to both terms, the GCF.<\/p>\n<div id=\"fs-id2502503\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>5<\/mn><msup>\n<mi>x<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>80<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>5<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>(<\/mo><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>16<\/mn><mo>)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>5<\/mn><msup>\n<mi>x<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>80<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>5<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>(<\/mo><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>16<\/mn><mo>)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1924982\">Notice that we have the difference of squares in the factor<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>16<\/mn><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>16<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>which we will continue to factor and obtain two solutions. The first term,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>5<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>5<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>generates, technically, two solutions as the exponent is 2, but they are the same solution.<\/p>\n<div id=\"fs-id2824827\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>5<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>16<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>4<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22124<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>5<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>16<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>4<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22124<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2638784\">The solutions are<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>0<\/mn><mspace width=\"0.5em\"><\/mspace><mtext>(double\u00a0solution),<\/mtext>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>0<\/mn><mspace width=\"0.5em\"><\/mspace><mtext>(double\u00a0solution),<\/mtext>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>4<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>4<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>\u22124.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22124.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/details>\n<div id=\"fs-id1693367\" data-type=\"commentary\">\n<h3 data-type=\"commentary-title\"><span class=\"os-title-label\">Analysis <\/span><\/h3>\n<p id=\"fs-id768264\">We can see the solutions on the graph in <a class=\"autogenerated-content\" href=\"2-6-other-types-of-equations#Figure_02_06_001\">Figure 1<\/a>. The <em data-effect=\"italics\">x-<\/em>coordinates of the points where the graph crosses the <em data-effect=\"italics\">x-<\/em>axis are the solutions\u2014the <em data-effect=\"italics\">x-<\/em>intercepts. Notice on the graph that at the solution<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>0<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>0<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>the graph touches the <em data-effect=\"italics\">x-<\/em>axis and bounces back. It does not cross the <em data-effect=\"italics\">x-<\/em>axis. This is typical of double solutions.<\/p>\n<div id=\"Figure_02_06_001\" class=\"os-figure\">\n<figure class=\"small\" data-id=\"Figure_02_06_001\"><span id=\"fs-id1548359\" data-type=\"media\" data-alt=\"Coordinate plane with the x-axis ranging from negative 5 to 5 and the y-axis ranging from negative 400 to 500 in intervals of 100. The function five times x to the fourth power minus eighty x squared equals zero is graphed along with the points (negative 4,0), (0,0), and (4,0).\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Coordinate plane with the x-axis ranging from negative 5 to 5 and the y-axis ranging from negative 400 to 500 in intervals of 100. The function five times x to the fourth power minus eighty x squared equals zero is graphed along with the points (negative 4,0), (0,0), and (4,0).\" width=\"487\" height=\"401\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/2e49a8dfcd752028a5ab6adfd4ad7bfdcb6e68a4\" \/><br \/>\n<\/span><\/figure>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span><br \/>\n<span class=\"os-number\">1<\/span><\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id988839\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#4<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_06_04\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2496351\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id3040346\">Solve by factoring:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>12<\/mn><msup>\n<mi>x<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>12<\/mn><msup>\n<mi>x<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_06_05\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">5<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id2435055\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1007381\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solve a Polynomial by Grouping<\/h3>\n<p id=\"fs-id2501656\">Solve a polynomial by grouping:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msup>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>+<\/mo><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>9<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>9<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>+<\/mo><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>9<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>9<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1475994\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1422651\">This polynomial consists of 4 terms, which we can solve by grouping. Grouping procedures require factoring the first two terms and then factoring the last two terms. If the factors in the parentheses are identical, we can continue the process and solve, unless more factoring is suggested.<\/p>\n<div id=\"fs-id2516826\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>+<\/mo><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>9<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>9<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>9<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mo>(<\/mo><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>9<\/mn><mo>)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>+<\/mo><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>9<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>9<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>9<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mo>(<\/mo><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>9<\/mn><mo>)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1408391\">The grouping process ends here, as we can factor<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>9<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>9<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><br \/>\nusing the difference of squares formula.<\/p>\n<div id=\"fs-id1402146\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd>\n<mrow>\n<mo>(<\/mo><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>9<\/mn><mo>)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>3<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22123<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22121<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd>\n<mrow>\n<mo>(<\/mo><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>9<\/mn><mo>)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>3<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22123<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22121<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2528732\">The solutions are<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22123<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22123<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>\u22121.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22121.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>Note that the highest exponent is 3 and we obtained 3 solutions. We can see the solutions, the <em data-effect=\"italics\">x-<\/em>intercepts, on the graph in <a class=\"autogenerated-content\" href=\"2-6-other-types-of-equations#Figure_02_06_002\">Figure 2<\/a><strong>.<\/strong><\/p>\n<div id=\"Figure_02_06_002\" class=\"os-figure\">\n<figure class=\"small\" data-id=\"Figure_02_06_002\"><span id=\"fs-id2304259\" data-type=\"media\" data-alt=\"Coordinate plane with the x-axis ranging from negative 5 to 5 and the y-axis ranging from negative 30 to 20 in intervals of 5. The function x cubed plus x squared minus nine times x minus nine equals zero is graphed along with the points (negative 3,0), (negative 1,0), and (3,0).\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Coordinate plane with the x-axis ranging from negative 5 to 5 and the y-axis ranging from negative 30 to 20 in intervals of 5. The function x cubed plus x squared minus nine times x minus nine equals zero is graphed along with the points (negative 3,0), (negative 1,0), and (3,0).\" width=\"402\" height=\"370\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/94b4e297364251745bf2d31918af414b117764c9\" \/><br \/>\n<\/span><\/figure>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span><br \/>\n<span class=\"os-number\">2<\/span><\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/details>\n<div id=\"fs-id2504109\" data-type=\"commentary\">\n<h3 data-type=\"commentary-title\"><span class=\"os-title-label\">Analysis <\/span><\/h3>\n<p id=\"fs-id1243628\">We looked at solving quadratic equations by factoring when the leading coefficient is 1. When the leading coefficient is not 1, we solved by grouping. Grouping requires four terms, which we obtained by splitting the linear term of quadratic equations. We can also use grouping for some polynomials of degree higher than 2, as we saw here, since there were already four terms.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2506398\" data-depth=\"1\">\n<h2 data-type=\"title\">Solving Radical Equations<\/h2>\n<p id=\"fs-id1846458\"><strong>Radical equations<\/strong> are equations that contain variables in the <span id=\"term-00002\" class=\"no-emphasis\" data-type=\"term\">radicand<\/span> (the expression under a radical symbol), such as<\/p>\n<div id=\"fs-id2781660\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>18<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mi>x<\/mi>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><\/mrow>\n<\/msqrt>\n<mo>\u2212<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>2<\/mn>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>18<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mi>x<\/mi>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><\/mrow>\n<\/msqrt>\n<mo>\u2212<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>2<\/mn>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1719179\">Radical equations may have one or more radical terms, and are solved by eliminating each radical, one at a time. We have to be careful when solving radical equations, as it is not unusual to find <span id=\"term-00003\" data-type=\"term\">extraneous solutions<\/span>, roots that are not, in fact, solutions to the equation. These solutions are not due to a mistake in the solving method, but result from the process of raising both sides of an equation to a power. However, checking each answer in the original equation will confirm the true solutions.<\/p>\n<div id=\"fs-id2506978\" class=\"ui-has-child-title\" data-type=\"note\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Radical Equations<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"eip-id3052762\">An equation containing terms with a variable in the radicand is called a <span id=\"term-00004\" data-type=\"term\">radical equation<\/span>.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2307719\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1517012\"><strong>Given a radical equation, solve it.<\/strong><\/p>\n<ol id=\"fs-id3143660\" type=\"1\">\n<li>Isolate the radical expression on one side of the equal sign. Put all remaining terms on the other side.<\/li>\n<li>If the radical is a square root, then square both sides of the equation. If it is a cube root, then raise both sides of the equation to the third power. In other words, for an <em data-effect=\"italics\">n<\/em>th root radical, raise both sides to the <em data-effect=\"italics\">n<\/em>th power. Doing so eliminates the radical symbol.<\/li>\n<li>Solve the remaining equation.<\/li>\n<li>If a radical term still remains, repeat steps 1\u20132.<\/li>\n<li>Confirm solutions by substituting them into the original equation.<\/li>\n<\/ol>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_06_06\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">6<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1222816\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1551039\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving an Equation with One Radical<\/h3>\n<p id=\"fs-id1520471\">Solve<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msqrt>\n<mrow>\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mi>x<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msqrt>\n<mrow>\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mi>x<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id2381217\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1332305\">The radical is already isolated on the left side of the equal side, so proceed to square both sides.<\/p>\n<div id=\"fs-id1919300\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mi>x<\/mi>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msqrt>\n<mrow>\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mi>x<\/mi>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msqrt>\n<mrow>\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1895386\">We see that the remaining equation is a quadratic. Set it equal to zero and solve.<\/p>\n<div id=\"fs-id1905278\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mn>0<\/mn>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22125<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>3<\/mn>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mn>0<\/mn>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>15<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22125<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>3<\/mn>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1257720\">The proposed solutions are<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22125<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22125<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>Let us check each solution back in the original equation. First, check<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>x<\/mi><mo>=<\/mo><mn>\u22125.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>=<\/mo><mn>\u22125.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id1463741\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mi>x<\/mi>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22125<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>25<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22125<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mn>5<\/mn>\n<\/mtd>\n<mtd>\n<mo>\u2260<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo><\/mo><mn>\u22125<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mi>x<\/mi>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mo>\u2212<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22125<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>25<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22125<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mn>5<\/mn>\n<\/mtd>\n<mtd>\n<mo>\u2260<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo><\/mo><mn>\u22125<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2628354\">This is an extraneous solution. While no mistake was made solving the equation, we found a solution that does not satisfy the original equation.<\/p>\n<p id=\"fs-id1417225\">Check<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>x<\/mi><mo>=<\/mo><mn>3.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>=<\/mo><mn>3.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id1515949\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mi>x<\/mi>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>3<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msqrt>\n<mn>9<\/mn>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>3<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mn>3<\/mn>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>3<\/mn>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mi>x<\/mi>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>15<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>3<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msqrt>\n<mn>9<\/mn>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>3<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mn>3<\/mn>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>3<\/mn>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1199291\">The solution is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>3.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>3.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1441693\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#5<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_06_05\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1583879\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1583880\">Solve the radical equation:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msqrt>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msqrt>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_06_07\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">7<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1720874\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2499586\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving a Radical Equation Containing Two Radicals<\/h3>\n<p id=\"fs-id1410869\">Solve<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msqrt>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>+<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mn>4.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msqrt>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>+<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mn>4.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1719019\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1929508\">As this equation contains two radicals, we isolate one radical, eliminate it, and then isolate the second radical.<\/p>\n<div id=\"fs-id2268062\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<mo>+<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>4<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Subtract\u00a0<\/mtext><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/msqrt>\n<mspace width=\"0.5em\"><\/mspace><mtext>from\u00a0both\u00a0sides<\/mtext><mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msqrt>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mrow>\n<mo>(<\/mo><mn>4<\/mn><mo>\u2212<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/msqrt>\n<mo>)<\/mo><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Square\u00a0both\u00a0sides<\/mtext><mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<mo>+<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>4<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Subtract\u00a0<\/mtext><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/msqrt>\n<mspace width=\"0.5em\"><\/mspace><mtext>from\u00a0both\u00a0sides<\/mtext><mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msqrt>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<msup>\n<mrow>\n<mo>(<\/mo><mn>4<\/mn><mo>\u2212<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/msqrt>\n<mo>)<\/mo><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Square\u00a0both\u00a0sides<\/mtext><mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1548786\">Use the perfect square formula to expand the right side:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mn>\u22122<\/mn><mi>a<\/mi><mi>b<\/mi><mo>+<\/mo><msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>a<\/mi><mo>\u2212<\/mo><mi>b<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mn>\u22122<\/mn><mi>a<\/mi><mi>b<\/mi><mo>+<\/mo><msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id2506054\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>+<\/mo><msup>\n<mrow>\n<mo>(<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>)<\/mo>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>16<\/mn><mo>\u2212<\/mo><mn>8<\/mn><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>14<\/mn><mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>8<\/mn><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Combine\u00a0like\u00a0terms<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>11<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>\u22128<\/mn><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Isolate\u00a0the\u00a0second\u00a0radical<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>11<\/mn><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msup>\n<mrow>\n<mo>(<\/mo><mn>\u22128<\/mn><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>)<\/mo>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Square\u00a0both\u00a0sides<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>22<\/mn><mi>x<\/mi><mo>+<\/mo><mn>121<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>64<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>+<\/mo><msup>\n<mrow>\n<mo>(<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>)<\/mo>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>16<\/mn><mo>\u2212<\/mo><mn>8<\/mn><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>+<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>14<\/mn><mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>8<\/mn><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Combine\u00a0like\u00a0terms<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>11<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>\u22128<\/mn><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Isolate\u00a0the\u00a0second\u00a0radical<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>11<\/mn><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msup>\n<mrow>\n<mo>(<\/mo><mn>\u22128<\/mn><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>)<\/mo>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Square\u00a0both\u00a0sides<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>22<\/mn><mi>x<\/mi><mo>+<\/mo><mn>121<\/mn>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>64<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1175462\">Now that both radicals have been eliminated, set the quadratic equal to zero and solve.<\/p>\n<div id=\"fs-id2946619\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>22<\/mn><mi>x<\/mi><mo>+<\/mo><mn>121<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>64<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>128<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>86<\/mn><mi>x<\/mi><mo>+<\/mo><mn>249<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>83<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Factor\u00a0and\u00a0solve<\/mtext><mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>3<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>83<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>22<\/mn><mi>x<\/mi><mo>+<\/mo><mn>121<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>64<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>128<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>86<\/mn><mi>x<\/mi><mo>+<\/mo><mn>249<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>83<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>Factor\u00a0and\u00a0solve<\/mtext><mo>.<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>3<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>83<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1563543\">The proposed solutions are<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>3<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>3<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>83.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>83.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>Check each solution in the original equation.<\/p>\n<div id=\"fs-id2655511\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<mo>+<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>4<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msqrt>\n<mn>9<\/mn>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\n<mn>1<\/mn>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mn>3<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>3<\/mn>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<mo>+<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>4<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msqrt>\n<mn>9<\/mn>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\n<mn>1<\/mn>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mn>3<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>3<\/mn>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1840544\">One solution is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>3.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>3.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p id=\"fs-id1918601\">Check<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>x<\/mi><mo>=<\/mo><mn>83.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>x<\/mi><mo>=<\/mo><mn>83.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id1941177\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<mo>+<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>4<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>83<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>83<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>169<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\n<mrow>\n<mn>81<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>13<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>\u2260<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<mo>+<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>4<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>83<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>83<\/mn><mo>\u2212<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>169<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>4<\/mn><mo>\u2212<\/mo><msqrt>\n<mrow>\n<mn>81<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>13<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>\u2260<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2293780\">The only solution is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>3.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>3.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>We see that<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>83<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>83<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is an extraneous solution.<\/p>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2430499\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#6<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_06_06\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2702970\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id2429740\">Solve the equation with two radicals:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msqrt>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>+<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mn>1.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msqrt>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>+<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mn>1.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id3240737\" data-depth=\"1\">\n<h2 data-type=\"title\">Solving an Absolute Value Equation<\/h2>\n<p id=\"fs-id1335626\">Next, we will learn how to solve an <span id=\"term-00005\" class=\"no-emphasis\" data-type=\"term\">absolute value equation<\/span>. To solve an equation such as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>|<\/mo> <mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>8<\/mn><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>8<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>we notice that the absolute value will be equal to 8 if the quantity inside the absolute value bars is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>8<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>8<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>or<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>\u22128.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u22128.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>This leads to two different equations we can solve independently.<\/p>\n<div id=\"fs-id2638694\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>8<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>or<\/mtext><mspace width=\"2em\"><\/mspace><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>\u22128<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mrow>\n<mn>14<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>\u22122<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>7<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>\u22121<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>8<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mrow>\n<mspace width=\"2em\"><\/mspace><mtext>or<\/mtext><mspace width=\"2em\"><\/mspace><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>\u22128<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mrow>\n<mn>14<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>\u22122<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mn>7<\/mn>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>\u22121<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1280820\">Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.<\/p>\n<div id=\"fs-id2918915\" class=\"ui-has-child-title\" data-type=\"note\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Absolute Value Equations<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1542481\">The absolute value of <em data-effect=\"italics\">x <\/em>is written as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>|<\/mo> <mi>x<\/mi> <mo>|<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mi>x<\/mi> <mo>|<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>It has the following properties:<\/p>\n<div id=\"fs-id2434805\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mtext>If\u00a0<\/mtext><mi>x<\/mi><mo>\u2265<\/mo><mn>0<\/mn><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mtext>then\u00a0<\/mtext><mo>|<\/mo><mi>x<\/mi><mo>|<\/mo><mo>=<\/mo><mi>x<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mtext>If\u00a0<\/mtext><mi>x<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mtext>then\u00a0<\/mtext><mo>|<\/mo><mi>x<\/mi><mo>|<\/mo><mo>=<\/mo><mi>\u2212<\/mi><mi>x<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mtext>If\u00a0<\/mtext><mi>x<\/mi><mo>\u2265<\/mo><mn>0<\/mn><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mtext>then\u00a0<\/mtext><mo>|<\/mo><mi>x<\/mi><mo>|<\/mo><mo>=<\/mo><mi>x<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mtext>If\u00a0<\/mtext><mi>x<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mtext>then\u00a0<\/mtext><mo>|<\/mo><mi>x<\/mi><mo>|<\/mo><mo>=<\/mo><mi>\u2212<\/mi><mi>x<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2905392\">For real numbers<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>A<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>A<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>B<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>B<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>an equation of the form<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>|<\/mo> <mi>A<\/mi> <mo>|<\/mo><\/mrow><mo>=<\/mo><mi>B<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mi>A<\/mi> <mo>|<\/mo><\/mrow><mo>=<\/mo><mi>B<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>with<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>B<\/mi><mo>\u2265<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>B<\/mi><mo>\u2265<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>will have solutions when<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>A<\/mi><mo>=<\/mo><mi>B<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>A<\/mi><mo>=<\/mo><mi>B<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>or<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>A<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>B<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>A<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>B<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>If<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>B<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>B<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>the equation<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>|<\/mo> <mi>A<\/mi> <mo>|<\/mo><\/mrow><mo>=<\/mo><mi>B<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mi>A<\/mi> <mo>|<\/mo><\/mrow><mo>=<\/mo><mi>B<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>has no solution.<\/p>\n<p id=\"fs-id1408140\">An <span id=\"term-00006\" data-type=\"term\">absolute value equation<\/span> in the form<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>|<\/mo> <mrow>\n<mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mi>c<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mrow>\n<mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mi>c<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>has the following properties:<\/p>\n<div id=\"fs-id2655133\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mtext>If\u00a0\u00a0<\/mtext><mi>c<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo>,<\/mo><mo>|<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>|<\/mo><mo>=<\/mo><mi>c<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>has\u00a0no\u00a0solution<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mtext>If\u00a0\u00a0<\/mtext><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo><mo>|<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>|<\/mo><mo>=<\/mo><mi>c<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>has\u00a0one\u00a0solution<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mtext>If\u00a0\u00a0<\/mtext><mi>c<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo><mo>|<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>|<\/mo><mo>=<\/mo><mi>c<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>has\u00a0two\u00a0solutions<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mtext>If\u00a0\u00a0<\/mtext><mi>c<\/mi><mo>&lt;<\/mo><mn>0<\/mn><mo>,<\/mo><mo>|<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>|<\/mo><mo>=<\/mo><mi>c<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>has\u00a0no\u00a0solution<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mtext>If\u00a0\u00a0<\/mtext><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo><mo>|<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>|<\/mo><mo>=<\/mo><mi>c<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>has\u00a0one\u00a0solution<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mtext>If\u00a0\u00a0<\/mtext><mi>c<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo><mo>|<\/mo><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>|<\/mo><mo>=<\/mo><mi>c<\/mi><mspace width=\"0.5em\"><\/mspace><mtext>has\u00a0two\u00a0solutions<\/mtext><mo>.<\/mo>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3038796\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1793504\"><strong>Given an absolute value equation, solve it.<\/strong><\/p>\n<ol id=\"fs-id2938958\" type=\"1\">\n<li>Isolate the absolute value expression on one side of the equal sign.<\/li>\n<li>If<br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow><mi>c<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>c<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>write and solve two equations:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>=<\/mo><mi>c<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>=<\/mo><mi>c<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>c<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>a<\/mi><mi>x<\/mi><mo>+<\/mo><mi>b<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>c<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<\/ol>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_06_08\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">8<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id3105697\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1836188\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving Absolute Value Equations<\/h3>\n<p id=\"fs-id1836194\">Solve the following absolute value equations:<\/p>\n<ol id=\"fs-id2370595\" class=\"circled\" type=\"1\">\n<li><span class=\"token\">\u24d0<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>8<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>8<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li><span class=\"token\">\u24d1<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>\u22129<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>\u22129<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li><span class=\"token\">\u24d2<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>4<\/mn><mo>=<\/mo><mn>6<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>4<\/mn><mo>=<\/mo><mn>6<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li><span class=\"token\">\u24d3<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>\u22125<\/mn><mi>x<\/mi><mo>+<\/mo><mn>10<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>\u22125<\/mn><mi>x<\/mi><mo>+<\/mo><mn>10<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<details id=\"fs-id2683627\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<ol id=\"fs-id2683629\" class=\"circled\" type=\"1\">\n<li><span class=\"token\">\u24d0<\/span>\n<p id=\"fs-id1832720\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>8<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>8<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p id=\"fs-id2389886\">Write two equations and solve each:<\/p>\n<div id=\"eip-id1170402604810\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow><mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>8<\/mn>\n<\/mtd><\/mtr><\/mtable>\n&nbsp;\n<mtd columnalign=\"right\">\n<mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>\u22128<\/mn>\n<\/mtd>\n<mtr>\n<mtd columnalign=\"right\">\n<mn>6<\/mn><mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>4<\/mn>\n<\/mtd><\/mtr>\n&nbsp;\n<mtd columnalign=\"right\">\n<mn>6<\/mn><mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>\u221212<\/mn>\n<\/mtd>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mtd><\/mtr>\n&nbsp;\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>\u22122<\/mn>\n<\/mtd>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>8<\/mn>\n<\/mtd>&nbsp;\n<mtd columnalign=\"right\">\n<mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>\u22128<\/mn>\n<\/mtd>\n<\/mtr><mtr>\n<mtd columnalign=\"right\">\n<mn>6<\/mn><mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>4<\/mn>\n<\/mtd><\/mtr>\n&nbsp;\n<mtd columnalign=\"right\">\n<mn>6<\/mn><mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>\u221212<\/mn>\n<\/mtd>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mtd><\/mtr>\n&nbsp;\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>\u22122<\/mn>\n<\/mtd>\n<\/mtable><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id3203361\">The two solutions are<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22122.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22122.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/li>\n<li><span class=\"token\">\u24d1<\/span>\n<p id=\"fs-id1879343\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>\u22129<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>\u22129<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p id=\"fs-id2293707\">There is no solution as an absolute value cannot be negative.<\/p>\n<\/li>\n<li><span class=\"token\">\u24d2<\/span>\n<p id=\"fs-id2888792\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>4<\/mn><mo>=<\/mo><mn>6<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>4<\/mn><mo>=<\/mo><mn>6<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p id=\"fs-id2436076\">Isolate the absolute value expression and then write two equations.<\/p>\n<div id=\"fs-id3274908\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>6<\/mn><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow> <mo>|<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>10<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>10<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>\u221210<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>15<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>\u22125<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>5<\/mn><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>6<\/mn><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow> <mo>|<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>10<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>10<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>\u221210<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mrow><mn>15<\/mn><\/mrow><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>\u22125<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>5<\/mn><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2498828\">There are two solutions:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>5<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>5<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>5<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/li>\n<li><span class=\"token\">\u24d3<\/span>\n<p id=\"fs-id1340685\">(d)<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>\u22125<\/mn><mi>x<\/mi><mo>+<\/mo><mn>10<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>\u22125<\/mn><mi>x<\/mi><mo>+<\/mo><mn>10<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>0<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p id=\"fs-id2016618\">The equation is set equal to zero, so we have to write only one equation.<\/p>\n<div id=\"fs-id2431537\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>\u22125<\/mn><mi>x<\/mi><mo>+<\/mo><mn>10<\/mn><\/mrow>\n<\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>0<\/mn><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mn>\u22125<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mrow><mn>\u221210<\/mn><\/mrow><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>2<\/mn><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>\u22125<\/mn><mi>x<\/mi><mo>+<\/mo><mn>10<\/mn><\/mrow>\n<\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>0<\/mn><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mn>\u22125<\/mn><mi>x<\/mi><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mrow><mn>\u221210<\/mn><\/mrow><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>2<\/mn><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1351555\">There is one solution:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>2.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>2.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/li>\n<\/ol>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1535632\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#7<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_06_07\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2515512\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id2515513\">Solve the absolute value equation:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mo>|<\/mo>\n<mn>1<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi>\n<mo>|<\/mo><mo>+<\/mo><mn>8<\/mn><mo>=<\/mo><mn>13.<\/mn>&nbsp;\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>|<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><mo>|<\/mo><mo>+<\/mo><mn>8<\/mn><mo>=<\/mo><mn>13.<\/mn><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2653809\" data-depth=\"1\">\n<h2 data-type=\"title\">Solving Other Types of Equations<\/h2>\n<p id=\"fs-id3169548\">There are many other types of equations in addition to the ones we have discussed so far. We will see more of them throughout the text. Here, we will discuss equations that are in quadratic form, and rational equations that result in a quadratic.<\/p>\n<section id=\"fs-id1923869\" data-depth=\"2\">\n<h3 data-type=\"title\">Solving Equations in Quadratic Form<\/h3>\n<p id=\"fs-id1923873\"><strong>Equations in quadratic form <\/strong>are equations with three terms. The first term has a power other than 2. The middle term has an exponent that is one-half the exponent of the leading term. The third term is a constant. We can solve equations in this form as if they were quadratic. A few examples of these equations include<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msup>\n<mi>x<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>5<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo><msup>\n<mi>x<\/mi>\n<mn>6<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>7<\/mn><msup>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>8<\/mn><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\n<mi>x<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>5<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo><msup>\n<mi>x<\/mi>\n<mn>6<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>7<\/mn><msup>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>8<\/mn><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>In each one, doubling the exponent of the middle term equals the exponent on the leading term. We can solve these equations by substituting a variable for the middle term.<\/p>\n<div id=\"fs-id1834422\" class=\"ui-has-child-title\" data-type=\"note\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Quadratic Form<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id3094567\">If the exponent on the middle term is one-half of the exponent on the leading term, we have an <span id=\"term-00007\" data-type=\"term\">equation in quadratic form<\/span>, which we can solve as if it were a quadratic. We substitute a variable for the middle term to solve equations in quadratic form.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1954745\" class=\"how-to-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"How To\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"How To\"><span class=\"os-title-label\">How To<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1385726\"><strong>Given an equation quadratic in form, solve it.<\/strong><\/p>\n<ol id=\"fs-id1385730\" type=\"1\">\n<li>Identify the exponent on the leading term and determine whether it is double the exponent on the middle term.<\/li>\n<li>If it is, substitute a variable, such as <em data-effect=\"italics\">u<\/em>, for the variable portion of the middle term.<\/li>\n<li>Rewrite the equation so that it takes on the standard form of a quadratic.<\/li>\n<li>Solve using one of the usual methods for solving a quadratic.<\/li>\n<li>Replace the substitution variable with the original term.<\/li>\n<li>Solve the remaining equation.<\/li>\n<\/ol>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_06_09\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">9<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1357813\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1357816\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving a Fourth-degree Equation in Quadratic Form<\/h3>\n<p id=\"fs-id2486573\">Solve this fourth-degree equation:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1229226\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id1229229\">This equation fits the main criteria, that the power on the leading term is double the power on the middle term. Next, we will make a substitution for the variable term in the middle. Let<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>u<\/mi><mo>=<\/mo><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>u<\/mi><mo>=<\/mo><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>Rewrite the equation in <em data-effect=\"italics\">u<\/em>.<\/p>\n<div id=\"fs-id3155491\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><msup>\n<mi>u<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>2<\/mn><mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><msup>\n<mi>u<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>2<\/mn><mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2371956\">Now solve the quadratic.<\/p>\n<div id=\"fs-id2371960\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><msup>\n<mi>u<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>2<\/mn><mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>0<\/mn><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mi>u<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><msup>\n<mi>u<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>2<\/mn><mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>0<\/mn><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>3<\/mn><mi>u<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2308574\">Solve each factor and replace the original term for <em data-effect=\"italics\">u.<\/em><\/p>\n<div id=\"fs-id2797268\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><mi>u<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><mi>u<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22121<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>u<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u00b1<\/mo><mi>i<\/mi><msqrt>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><mi>u<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>3<\/mn><mi>u<\/mi><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22121<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>u<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u00b1<\/mo><mi>i<\/mi><msqrt>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<div id=\"fs-id1335933\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>u<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>1<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>1<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u00b11<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>u<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>1<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>1<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u00b11<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2980376\">The solutions are<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mo>\u00b1<\/mo><mi>i<\/mi><msqrt>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mo>\u00b1<\/mo><mi>i<\/mi><msqrt>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mo>\u00b1<\/mo><mn>1.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mo>\u00b1<\/mo><mn>1.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1929412\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#8<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_06_08\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2514218\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id2514219\">Solve using substitution:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msup>\n<mi>x<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>8<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>9<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\n<mi>x<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>8<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>9<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_06_10\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">10<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id2444359\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2444361\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving an Equation in Quadratic Form Containing a Binomial<\/h3>\n<p id=\"fs-id2029548\">Solve the equation in quadratic form:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>11<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mn>12<\/mn><mo>=<\/mo><mn>0.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>11<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mn>12<\/mn><mo>=<\/mo><mn>0.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id2382010\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id2382012\">This equation contains a binomial in place of the single variable. The tendency is to expand what is presented. However, recognizing that it fits the criteria for being in quadratic form makes all the difference in the solving process. First, make a substitution, letting<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>u<\/mi><mo>=<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>u<\/mi><mo>=<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>Then rewrite the equation in <em data-effect=\"italics\">u.<\/em><\/p>\n<div id=\"fs-id1929605\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msup>\n<mi>u<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>11<\/mn><mi>u<\/mi><mo>\u2212<\/mo><mn>12<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>u<\/mi><mo>+<\/mo><mn>12<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msup>\n<mi>u<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>11<\/mn><mi>u<\/mi><mo>\u2212<\/mo><mn>12<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>u<\/mi><mo>+<\/mo><mn>12<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1691070\">Solve using the zero-factor property and then replace <em data-effect=\"italics\">u<\/em> with the original expression.<\/p>\n<div id=\"fs-id1516932\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>u<\/mi><mo>+<\/mo><mn>12<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>u<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u221212<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u221212<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u221214<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>u<\/mi><mo>+<\/mo><mn>12<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>u<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u221212<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u221212<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u221214<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1720759\">The second factor results in<\/p>\n<div id=\"fs-id2485990\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>u<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>1<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>1<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22121<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>u<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>u<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>1<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>1<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22121<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1943470\">We have two solutions:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mn>\u221214<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mn>\u221214<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22121.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22121.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1736480\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#9<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_06_09\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1952007\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1952008\">Solve:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mn>21<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mn>21<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2893053\" data-depth=\"2\">\n<h3 data-type=\"title\">Solving Rational Equations Resulting in a Quadratic<\/h3>\n<p id=\"fs-id1679917\">Earlier, we solved rational equations. Sometimes, solving a rational equation results in a quadratic. When this happens, we continue the solution by simplifying the quadratic equation by one of the methods we have seen. It may turn out that there is no solution.<\/p>\n<div id=\"Example_02_06_11\" class=\"ui-has-child-title\" data-type=\"example\">\n<header>\n<h2 class=\"os-title\"><span class=\"os-title-label\">Example <\/span><br \/>\n<span class=\"os-number\">11<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id1970290\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2508561\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving a Rational Equation Leading to a Quadratic<\/h3>\n<p id=\"fs-id2508566\">Solve the following rational equation:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi>\n<\/mrow>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>4<\/mn>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>8<\/mn>\n<\/mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi>\n<\/mrow>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>4<\/mn>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>8<\/mn>\n<\/mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id2797285\" data-type=\"solution\" aria-label=\"Show\/Hide Solution\">\n<summary class=\"btn-link ui-toggle\" title=\"Show\/Hide Solution\" data-content=\"Show\/Hide Solution\"><\/summary>\n<section class=\"ui-body\" role=\"alert\">\n<h3 data-type=\"solution-title\"><span class=\"os-title-label\">Solution<\/span><\/h3>\n<div class=\"os-solution-container\">\n<p id=\"fs-id2797287\">We want all denominators in factored form to find the LCD. Two of the denominators cannot be factored further. However,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mn>\u22121<\/mn><mo>=<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mn>\u22121<\/mn><mo>=<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>Then, the LCD is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>Next, we multiply the whole equation by the LCD.<\/p>\n<div id=\"fs-id2441856\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mrow><mo>[<\/mo> <mrow>\n<mfrac>\n<mrow>\n<mn>\u22124<\/mn><mi>x<\/mi><\/mrow>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>4<\/mn>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<\/mfrac>\n<\/mrow> <mo>]<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mrow><mo>[<\/mo> <mrow>\n<mfrac>\n<mrow>\n<mn>\u22128<\/mn><\/mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mfrac>\n<\/mrow> <mo>]<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>\u22124<\/mn><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22128<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>\u22124<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22128<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>\u22124<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>\u22124<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>\u22124<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22121<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>1<\/mn>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mrow><mo>[<\/mo> <mrow>\n<mfrac>\n<mrow>\n<mn>\u22124<\/mn><mi>x<\/mi><\/mrow>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>4<\/mn>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<\/mfrac>\n<\/mrow> <mo>]<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mrow><mo>[<\/mo> <mrow>\n<mfrac>\n<mrow>\n<mn>\u22128<\/mn><\/mrow>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mfrac>\n<\/mrow> <mo>]<\/mo><\/mrow><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>\u22124<\/mn><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22128<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>\u22124<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22128<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>\u22124<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>\u22124<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>\u22124<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>0<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>\u22121<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mi>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>1<\/mn>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2980350\">In this case, either solution produces a zero in the denominator in the original equation. Thus, there is no solution.<\/p>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2906702\" class=\"precalculus try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Try It\">\n<header>\n<h2 class=\"os-title\" data-label-parent=\"Try It\"><span class=\"os-title-label\">Try It <\/span><br \/>\n<span class=\"os-number\">#10<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_02_06_10\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1751154\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1751155\">Solve<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mfrac>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>1<\/mn>\n<mi>x<\/mi>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mfrac>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mfrac>\n<mo>+<\/mo><mfrac>\n<mn>1<\/mn>\n<mi>x<\/mi>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>2<\/mn>\n<\/mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2665007\" class=\"media-notitle ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"Media\">\n<header>\n<h2 class=\"os-title\" data-type=\"title\" data-label-parent=\"Media\"><span class=\"os-title-label\">Media<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1822998\">Access these online resources for additional instruction and practice with different types of equations.<\/p>\n<ul id=\"fs-id1823002\">\n<li><a href=\"http:\/\/openstax.org\/l\/rateqnosoln\" target=\"_blank\" rel=\"noopener nofollow\">Rational Equation with no Solution<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/ratexprecpexp\" target=\"_blank\" rel=\"noopener nofollow\">Solving equations with rational exponents using reciprocal powers<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/radeqsolvepart1\" target=\"_blank\" rel=\"noopener nofollow\">Solving radical equations part 1 of 2<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/radeqsolvepart2\" target=\"_blank\" rel=\"noopener nofollow\">Solving radical equations part 2 of 2<\/a><\/li>\n<\/ul>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">2.6 Section Exercises<\/span><\/h2>\n<section id=\"fs-id3146179\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id3146185\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id2028956\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2028957\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2028956-solution\">1<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2028958\">In a radical equation, what does it mean if a number is an extraneous solution?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3256935\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3256936\" data-type=\"problem\"><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3256937\">Explain why possible solutions <em data-effect=\"italics\">must<\/em> be checked in radical equations.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2501234\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2501235\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2501234-solution\">3<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2501236\">Your friend tries to calculate the value<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mo>\u2212<\/mo><msup>\n<mn>9<\/mn>\n<mrow>\n<mstyle scriptlevel=\"+1\">\n<mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mstyle>&nbsp;\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mo>\u2212<\/mo><msup>\n<mn>9<\/mn>\n<mrow>\n<mstyle scriptlevel=\"+1\">\n<mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mstyle>&nbsp;\n<\/mrow>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>and keeps getting an ERROR message. What mistake are they probably making?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1941430\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1941431\" data-type=\"problem\"><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1941432\">Explain why<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mrow><mo>|<\/mo> <mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>\u22127<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mrow><mo>|<\/mo> <mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>\u22127<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>has no solutions.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1354283\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1354284\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1354283-solution\">5<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1354285\">Explain how to change a rational exponent into the correct radical expression.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id3018948\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<p id=\"fs-id2628823\">For the following exercises, solve the rational exponent equation. Use factoring where necessary.<\/p>\n<div id=\"fs-id2628826\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2628827\" data-type=\"problem\"><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2628828\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><mn>16<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><mn>16<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1973815\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1973816\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1973815-solution\">7<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1973817\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><mn>27<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><mn>27<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2432476\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2432477\" data-type=\"problem\"><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2432478\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3040226\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3040227\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3040226-solution\">9<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1296521\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><mn>8<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mrow>\n<mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><mn>8<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3060328\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3060329\" data-type=\"problem\"><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3060330\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><mn>4<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><mn>4<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1763592\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1763593\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1763592-solution\">11<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1763594\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><mn>5<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>+<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><mn>5<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>+<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1709842\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1709843\" data-type=\"problem\"><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1709844\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>4<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>7<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>4<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><msup>\n<mi>x<\/mi>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msup>\n<mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2382870\">For the following exercises, solve the following polynomial equations by grouping and factoring.<\/p>\n<div id=\"fs-id2382874\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2382875\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2382874-solution\">13<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2382876\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3111192\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3111193\" data-type=\"problem\"><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3111194\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>6<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>27<\/mn><mi>x<\/mi><mo>+<\/mo><mn>54<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>6<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>27<\/mn><mi>x<\/mi><mo>+<\/mo><mn>54<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1393262\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1393263\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1393262-solution\">15<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1393264\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>4<\/mn><msup>\n<mi>y<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>9<\/mn><mi>y<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>4<\/mn><msup>\n<mi>y<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>9<\/mn><mi>y<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2529330\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2529331\" data-type=\"problem\"><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2529332\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>25<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>75<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>25<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>75<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1582399\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1582400\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1582399-solution\">17<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1582402\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>m<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>+<\/mo><msup>\n<mi>m<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mi>m<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>m<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>+<\/mo><msup>\n<mi>m<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mi>m<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id586482\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id586483\" data-type=\"problem\"><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2998491\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>5<\/mn>\n<\/msup>\n<mn>\u221214<\/mn><msup>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>5<\/mn>\n<\/msup>\n<mn>\u221214<\/mn><msup>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>0<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2643100\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2643101\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2643100-solution\">19<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2643102\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>5<\/mn><msup>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>45<\/mn><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>18<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>5<\/mn><msup>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>45<\/mn><mi>x<\/mi><mo>=<\/mo><mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>18<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2917413\">For the following exercises, solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions.<\/p>\n<div id=\"fs-id2917418\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2294086\" data-type=\"problem\"><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2294087\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msqrt>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/msqrt>\n<mo>\u2212<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msqrt>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/msqrt>\n<mo>\u2212<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2332670\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2332671\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2332670-solution\">21<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2332672\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mn>5<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mn>5<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2375964\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2375966\" data-type=\"problem\"><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2375967\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msqrt>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2486495\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2486496\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2486495-solution\">23<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2486497\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msqrt>\n<mrow>\n<mn>3<\/mn><mi>t<\/mi><mo>+<\/mo><mn>5<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mn>7<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msqrt>\n<mrow>\n<mn>3<\/mn><mi>t<\/mi><mo>+<\/mo><mn>5<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mn>7<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1551969\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1551970\" data-type=\"problem\"><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1551971\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msqrt>\n<mrow>\n<mi>t<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>+<\/mo><mn>9<\/mn><mo>=<\/mo><mn>7<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msqrt>\n<mrow>\n<mi>t<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>+<\/mo><mn>9<\/mn><mo>=<\/mo><mn>7<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3256844\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3256845\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id3256844-solution\">25<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3256846\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msqrt>\n<mrow>\n<mn>12<\/mn><mo>\u2212<\/mo><mi>x<\/mi><\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mi>x<\/mi><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msqrt>\n<mrow>\n<mn>12<\/mn><mo>\u2212<\/mo><mi>x<\/mi><\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mi>x<\/mi><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1357710\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1357711\" data-type=\"problem\"><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1357712\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msqrt>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>\u2212<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msqrt>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>\u2212<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mn>2<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1441558\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1441559\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1441558-solution\">27<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1441560\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msqrt>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>+<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msqrt>\n<mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>+<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mn>1<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1951698\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1951699\" data-type=\"problem\"><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1951700\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msqrt>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<mo>\u2212<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mn>1<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msqrt>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<\/msqrt>\n<mo>\u2212<\/mo><msqrt>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mn>1<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1786258\">For the following exercises, solve the equation involving absolute value.<\/p>\n<div id=\"fs-id1150948\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1150949\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1150948-solution\">29<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1150950\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>8<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>8<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2307448\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2307449\" data-type=\"problem\"><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2307450\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>\u22122<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>\u22122<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2410873\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2410874\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2410873-solution\">31<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2410875\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>1<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>5<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>1<\/mn><mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>5<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2428664\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2428665\" data-type=\"problem\"><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2428666\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>3<\/mn><mo>=<\/mo><mn>6<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>3<\/mn><mo>=<\/mo><mn>6<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1717824\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1717825\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1717824-solution\">33<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1717826\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>7<\/mn><mo>=<\/mo><mn>\u22122<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>7<\/mn><mo>=<\/mo><mn>\u22122<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3165049\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3165050\" data-type=\"problem\"><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3165051\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>2<\/mn><mo>=<\/mo><mn>\u22123<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>\u2212<\/mo><mn>2<\/mn><mo>=<\/mo><mn>\u22123<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2382990\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2382991\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2382990-solution\">35<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2382992\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>0<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2643356\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2643357\" data-type=\"problem\"><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2643358\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo>\u2212<\/mo><mrow><mo>|<\/mo> <mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>\u22123<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>\u2212<\/mo><mrow><mo>|<\/mo> <mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>\u22123<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2665057\">For the following exercises, solve the equation by identifying the quadratic form. Use a substitute variable and find all real solutions by factoring.<\/p>\n<div id=\"fs-id2665062\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2665063\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2665062-solution\">37<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2386875\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>10<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>9<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>4<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>10<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>9<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1715518\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1715519\" data-type=\"problem\"><span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1715520\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>4<\/mn><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>t<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>9<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>t<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>\u22122<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>4<\/mn><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>t<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>9<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>t<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>\u22122<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1577029\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2029121\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1577029-solution\">39<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2029122\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mn>12<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mn>12<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1418519\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1418520\" data-type=\"problem\"><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1418521\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>8<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mn>9<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>8<\/mn><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>\u2212<\/mo><mn>9<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2797000\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2797001\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2797000-solution\">41<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2797002\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2390141\" data-depth=\"2\">\n<h3 data-type=\"title\">Extensions<\/h3>\n<p id=\"fs-id2390146\">For the following exercises, solve for the unknown variable.<\/p>\n<div id=\"fs-id2390150\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2390151\" data-type=\"problem\"><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2390152\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mn>\u22122<\/mn><\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><msup>\n<mi>x<\/mi>\n<mrow>\n<mn>\u22121<\/mn><\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><mn>12<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mrow>\n<mn>\u22122<\/mn><\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><msup>\n<mi>x<\/mi>\n<mrow>\n<mn>\u22121<\/mn><\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><mn>12<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1726737\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1726738\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1726737-solution\">43<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1726739\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msqrt>\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>|<\/mo> <mi>x<\/mi> <mo>|<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mi>x<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msqrt>\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>|<\/mo> <mi>x<\/mi> <mo>|<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/msqrt>\n<mo>=<\/mo><mi>x<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2412235\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2412236\" data-type=\"problem\"><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2412237\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>t<\/mi>\n<mrow>\n<mn>10<\/mn>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><mn>2<\/mn><msup>\n<mi>t<\/mi>\n<mn>5<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>t<\/mi>\n<mrow>\n<mn>10<\/mn>\n<\/mrow>\n<\/msup>\n<mo>\u2212<\/mo><mn>2<\/mn><msup>\n<mi>t<\/mi>\n<mn>5<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1276816\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1276817\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1276816-solution\">45<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1276818\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>|<\/mo> <mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>36<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>12<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>|<\/mo> <mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>36<\/mn>\n<\/mrow> <mo>|<\/mo><\/mrow><mo>=<\/mo><mn>12<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1976510\" data-depth=\"2\">\n<h3 data-type=\"title\">Real-World Applications<\/h3>\n<p id=\"fs-id1976516\">For the following exercises, use the model for the period of a pendulum,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>T<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>T<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>such that<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>T<\/mi><mo>=<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><msqrt>\n<mrow>\n<mfrac>\n<mi>L<\/mi>\n<mi>g<\/mi>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msqrt>\n<mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>T<\/mi><mo>=<\/mo><mn>2<\/mn><mi>\u03c0<\/mi><msqrt>\n<mrow>\n<mfrac>\n<mi>L<\/mi>\n<mi>g<\/mi>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msqrt>\n<mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>where the length of the pendulum is <em data-effect=\"italics\">L<\/em> and the acceleration due to gravity is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>g<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>g<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id2500088\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2500089\" data-type=\"problem\"><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2500090\">If the acceleration due to gravity is 9.8 m\/s<sup>2<\/sup> and the period equals 1 s, find the length to the nearest cm (100 cm = 1 m).<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2947584\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2947585\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id2947584-solution\">47<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1762772\">If the gravity is 32 ft\/s<sup>2<\/sup> and the period equals 1 s, find the length to the nearest in. (12 in. = 1 ft). Round your answer to the nearest in.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1591913\">For the following exercises, use a model for body surface area, BSA, such that<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>B<\/mi><mi>S<\/mi><mi>A<\/mi><mo>=<\/mo><msqrt>\n<mrow>\n<mfrac>\n<mrow>\n<mi>w<\/mi><mi>h<\/mi>\n<\/mrow>\n<mrow>\n<mn>3600<\/mn>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msqrt>\n<mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>B<\/mi><mi>S<\/mi><mi>A<\/mi><mo>=<\/mo><msqrt>\n<mrow>\n<mfrac>\n<mrow>\n<mi>w<\/mi><mi>h<\/mi>\n<\/mrow>\n<mrow>\n<mn>3600<\/mn>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/msqrt>\n<mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>where <em data-effect=\"italics\">w<\/em> = weight in kg and <em data-effect=\"italics\">h<\/em> = height in cm.<\/p>\n<div id=\"fs-id3070467\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3070468\" data-type=\"problem\"><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3070470\">Find the height of a 72-kg female to the nearest cm whose<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>B<\/mi><mi>S<\/mi><mi>A<\/mi><mo>=<\/mo><mn>1.8.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>B<\/mi><mi>S<\/mi><mi>A<\/mi><mo>=<\/mo><mn>1.8.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1591172\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1591174\" data-type=\"problem\"><a class=\"os-number\" href=\"chapter-2\" data-page-slug=\"chapter-2\" data-page-uuid=\"56cb9f76-1e9d-5e02-b989-634734fbbd83\" data-page-fragment=\"fs-id1591172-solution\">49<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1591175\">Find the weight of a 177-cm male to the nearest kg whose<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>B<\/mi><mi>S<\/mi><mi>A<\/mi><mo>=<\/mo><mn>2.1.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>B<\/mi><mi>S<\/mi><mi>A<\/mi><mo>=<\/mo><mn>2.1.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":158,"menu_order":6,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-112","chapter","type-chapter","status-publish","hentry"],"part":51,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/112","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/users\/158"}],"version-history":[{"count":2,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/112\/revisions"}],"predecessor-version":[{"id":323,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/112\/revisions\/323"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/parts\/51"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/112\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=112"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=112"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=112"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=112"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}