{"id":111,"date":"2025-04-09T17:10:16","date_gmt":"2025-04-09T17:10:16","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/2-5-quadratic-equations-college-algebra-2e-openstax\/"},"modified":"2026-03-19T15:56:32","modified_gmt":"2026-03-19T15:56:32","slug":"2-5-quadratic-equations","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/2-5-quadratic-equations\/","title":{"raw":"2.5 Quadratic Equations","rendered":"2.5 Quadratic 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_00138001-23fe-4ec9-b2c6-000f3f28ee23\" 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 quadratic equations by factoring.<\/li>\r\n \t<li>Solve quadratic equations by the square root property.<\/li>\r\n \t<li>Solve quadratic equations by completing the square.<\/li>\r\n \t<li>Solve quadratic equations by using the quadratic formula.<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<div id=\"Figure_02_05_001\" class=\"os-figure\">\r\n<figure class=\"medium\" data-id=\"Figure_02_05_001\"><span id=\"fs-id1278656\" data-type=\"media\" data-alt=\"Two televisions side-by-side. The right television is slightly larger than the left.\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Two televisions side-by-side. The right television is slightly larger than the left.\" width=\"655\" height=\"273\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/fa20f121f369fdc2ecdc21a016fa21fe596806bf\" \/>\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<p id=\"fs-id1087569\">The computer monitor on the left in <a class=\"autogenerated-content\" href=\"2-5-quadratic-equations#Figure_02_05_001\">Figure 1<\/a> is a 23.6-inch model and the one on the right is a 27-inch model. Proportionally, the monitors appear very similar. If there is a limited amount of space and we desire the largest monitor possible, how do we decide which one to choose? In this section, we will learn how to solve problems such as this using four different methods.<\/p>\r\n\r\n<section id=\"fs-id3182628\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Solving Quadratic Equations by Factoring<\/h2>\r\n<p id=\"fs-id2980280\">An equation containing a second-degree polynomial is called a <span id=\"term-00001\" class=\"no-emphasis\" data-type=\"term\">quadratic equation<\/span>. For example, equations such as<\/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>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/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>2<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><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>4<\/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>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>are quadratic equations. They are used in countless ways in the fields of engineering, architecture, finance, biological science, and, of course, mathematics.\r\n<p id=\"fs-id1385706\">Often the easiest method of solving a quadratic equation is <span id=\"term-00002\" class=\"no-emphasis\" data-type=\"term\">factoring<\/span>. Factoring means finding expressions that can be multiplied together to give the expression on one side of the equation.<\/p>\r\n<p id=\"fs-id2521760\">If a quadratic equation can be factored, it is written as a product of linear terms. Solving by factoring depends on the zero-product property, which states that if<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u22c5<\/mo><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>\u22c5<\/mo><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>then\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>or\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>b<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>b<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>where <em data-effect=\"italics\">a <\/em>and <em data-effect=\"italics\">b <\/em>are real numbers or algebraic expressions. In other words, if the product of two numbers or two expressions equals zero, then one of the numbers or one of the expressions must equal zero because zero multiplied by anything equals zero.\r\n<p id=\"fs-id1338084\">Multiplying the factors expands the equation to a string of terms separated by plus or minus signs. So, in that sense, the operation of multiplication undoes the operation of factoring. For example, expand the factored expression<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>by multiplying the two factors together.\r\n<div id=\"fs-id2921580\" 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<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/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<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/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<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/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<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/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<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\"><\/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<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/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-id1280231\">The product is a quadratic expression. Set equal to zero,<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is a quadratic equation. If we were to factor the equation, we would get back the factors we multiplied.\r\n<p id=\"fs-id2996324\">The process of factoring a quadratic equation depends on the leading coefficient, whether it is 1 or another integer. We will look at both situations; but first, we want to confirm that the equation is written in standard form,<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>where <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are real numbers, and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u2260<\/mo><mn>0.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>\u2260<\/mo><mn>0.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>The equation\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/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<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>is in standard form.\r\n<p id=\"fs-id3176714\">We can use the zero-product property to solve quadratic equations in which we first have to factor out the <span id=\"term-00003\" class=\"no-emphasis\" data-type=\"term\">greatest common factor<\/span> (GCF), and for equations that have special factoring formulas as well, such as the difference of squares, both of which we will see later in this section.<\/p>\r\n\r\n<div id=\"fs-id1318343\" 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=\"\">The Zero-Product Property and Quadratic Equations <\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id2437408\">The <span id=\"term-00004\" data-type=\"term\">zero-product property<\/span> states<\/p>\r\n\r\n<div id=\"fs-id3130490\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtext>If\u00a0<\/mtext><mi>a<\/mi><mo>\u22c5<\/mo><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mtext>then\u00a0<\/mtext><mi>a<\/mi><mo>=<\/mo><mn>0<\/mn><mspace width=\"0.5em\"><\/mspace><mtext>or\u00a0<\/mtext><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtext>If\u00a0<\/mtext><mi>a<\/mi><mo>\u22c5<\/mo><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mtext>then\u00a0<\/mtext><mi>a<\/mi><mo>=<\/mo><mn>0<\/mn><mspace width=\"0.5em\"><\/mspace><mtext>or\u00a0<\/mtext><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2001660\">where <em data-effect=\"italics\">a <\/em>and <em data-effect=\"italics\">b <\/em>are real numbers or algebraic expressions.<\/p>\r\n<p id=\"fs-id1297051\">A <span id=\"term-00005\" data-type=\"term\">quadratic equation<\/span> is an equation containing a second-degree polynomial; for example<\/p>\r\n\r\n<div id=\"fs-id2714980\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1274659\">where <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are real numbers, and if<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u2260<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>\u2260<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>it is in standard form.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1752173\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Solving Quadratics with a Leading Coefficient of 1<\/h3>\r\n<p id=\"fs-id2049812\">In the quadratic equation<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>the leading coefficient, or the coefficient of\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<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>\r\n<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>is 1. We have one method of factoring quadratic equations in this form.\r\n<div id=\"fs-id2048827\" 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-id1279785\"><strong>Given a quadratic equation with the leading coefficient of 1, factor it.<\/strong><\/p>\r\n\r\n<ol id=\"fs-id2436584\" type=\"1\">\r\n \t<li>Find two numbers whose product equals <em data-effect=\"italics\">c<\/em> and whose sum equals <em data-effect=\"italics\">b<\/em>.<\/li>\r\n \t<li>Use those numbers to write two factors of the form\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mi>k<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mspace width=\"0.5em\"><\/mspace><mtext>or\u00a0<\/mtext><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mi>k<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mi>k<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mspace width=\"0.5em\"><\/mspace><mtext>or\u00a0<\/mtext><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mi>k<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>where <em data-effect=\"italics\">k <\/em>is one of the numbers found in step 1. Use the numbers exactly as they are. In other words, if the two numbers are 1 and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22122<\/mn><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22122<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>the factors are\r\n\r\n<math display=\"inline\"><semantics><mrow>\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><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><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><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li>Solve using the zero-product property by setting each factor equal to zero and solving for the variable.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_05_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-id1315053\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2697997\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Factoring and Solving a Quadratic with Leading Coefficient of 1<\/h3>\r\n<p id=\"fs-id1207622\">Factor and solve the equation:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/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<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/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<details id=\"fs-id2508977\" 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-id1815377\">To factor<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>we look for two numbers whose product equals\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22126<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22126<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and whose sum equals 1. Begin by looking at the possible factors of\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22126.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22126.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id2771168\" 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<mn>1<\/mn><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22126<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>\u22126<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd>\r\n<mrow>\r\n<mn>2<\/mn><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22123<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd>\r\n<mrow>\r\n<mn>3<\/mn><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">)<\/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>\r\n<mtd>\r\n<mrow>\r\n<mn>1<\/mn><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22126<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>\u22126<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd>\r\n<mrow>\r\n<mn>2<\/mn><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22123<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd>\r\n<mrow>\r\n<mn>3<\/mn><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">)<\/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-id1722929\">The last pair,<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><mo>\u22c5<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22122<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><mo>\u22c5<\/mo><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22122<\/mn>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>sums to 1, so these are the numbers. Note that only one pair of numbers will work. Then, write the factors.\r\n<div id=\"fs-id1466455\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1779896\">To solve this equation, we use the zero-product property. Set each factor equal to zero and solve.<\/p>\r\n\r\n<div id=\"fs-id2437878\" 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<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" 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<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/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\"><mi>x<\/mi><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mn>2<\/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<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/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\"><mi>x<\/mi><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mrow><mn>\u22123<\/mn><\/mrow><\/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<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd rowalign=\"center\" 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<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/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\"><mi>x<\/mi><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mn>2<\/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<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/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\"><mi>x<\/mi><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mrow><mn>\u22123<\/mn><\/mrow><\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2753851\">The two solutions are<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22123.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22123.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>We can see how the solutions relate to the graph in <a class=\"autogenerated-content\" href=\"2-5-quadratic-equations#Figure_02_05_002\">Figure 2<\/a>. The solutions are the <em data-effect=\"italics\">x-<\/em>intercepts of\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow><mi>y<\/mi>\r\n<mo>=<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0.<\/mn>\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>y<\/mi>\r\n<mo>=<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0.<\/mn>\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"Figure_02_05_002\" class=\"os-figure\">\r\n<figure class=\"small\" data-id=\"Figure_02_05_002\"><span id=\"fs-id2381086\" data-type=\"media\" data-alt=\"Coordinate plane with the x-axis ranging from negative 5 to 5 and the y-axis ranging from negative 7 to 7. The function x squared plus x minus six equals zero is graphed, with the x-intercepts (-3,0) and (2,0), plotted as well.\">\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 7 to 7. The function x squared plus x minus six equals zero is graphed, with the x-intercepts (-3,0) and (2,0), plotted as well.\" width=\"487\" height=\"588\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/d245f4bb5c2445c5f7b842f335d44530209b28d6\" \/>\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><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1510352\" 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_05_01\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1769680\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2294713\">Factor and solve the quadratic equation:<\/p>\r\n\r\n<math display=\"inline\"><semantics><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>5<\/mn><mi>x<\/mi><mo>\u2212<\/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<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/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>\r\n<\/section><\/div>\r\n<div id=\"Example_02_05_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-id3052323\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1996793\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solve the Quadratic Equation by Factoring<\/h3>\r\n<p id=\"fs-id1045593\">Solve the quadratic equation by factoring:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>8<\/mn><mi>x<\/mi><mo>+<\/mo><mn>15<\/mn><mo>=<\/mo><mn>0.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>8<\/mn><mi>x<\/mi><mo>+<\/mo><mn>15<\/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-id1834290\" 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-id1690316\">Find two numbers whose product equals<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>15<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>15<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and whose sum equals\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>8.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>8.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>List the factors of\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>15.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>15.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id2892822\" 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=\"left\"><mrow><mn>1<\/mn><mo>\u22c5<\/mo><mn>15<\/mn><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"left\"><mrow><mn>3<\/mn><mo>\u22c5<\/mo><mn>5<\/mn><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"left\"><mrow><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u221215<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"left\"><mrow><mo stretchy=\"false\">(<\/mo><mn>\u22123<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22125<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/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=\"left\"><mrow><mn>1<\/mn><mo>\u22c5<\/mo><mn>15<\/mn><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"left\"><mrow><mn>3<\/mn><mo>\u22c5<\/mo><mn>5<\/mn><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"left\"><mrow><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u221215<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"left\"><mrow><mo stretchy=\"false\">(<\/mo><mn>\u22123<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22125<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2629248\">The numbers that add to 8 are 3 and 5. Then, write the factors, set each factor equal to zero, and solve.<\/p>\r\n\r\n<div id=\"fs-id2364916\" 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<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/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<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/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<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<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/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<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>&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<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/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<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/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<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<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/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<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>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1288611\">The solutions are<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22123<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22123<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\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>&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2431514\" 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_05_02\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1515965\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1545876\">Solve the quadratic equation by factoring:<\/p>\r\n\r\n<math display=\"inline\"><semantics><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>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>21<\/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>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><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<div id=\"Example_02_05_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-id1442204\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1766773\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Using the Zero-Product Property to Solve a Quadratic Equation Written as the Difference of Squares<\/h3>\r\n<p id=\"fs-id1007551\">Solve the difference of squares equation using the zero-product property:<\/p>\r\n\r\n<math display=\"inline\"><semantics><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>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>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<details id=\"fs-id1467645\" 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-id1342314\">Recognizing that the equation represents the difference of squares, we can write the two factors by taking the square root of each term, using a minus sign as the operator in one factor and a plus sign as the operator in the other. Solve using the zero-factor property.<\/p>\r\n\r\n<div id=\"fs-id1723089\" 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\"><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>9<\/mn><\/mrow><\/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<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/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><mrow><mo>(<\/mo><mrow><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow><mo>)<\/mo><\/mrow><\/mrow><\/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\"><mi>x<\/mi><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mn>3<\/mn><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mrow><mo>(<\/mo><mrow><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow><mo>)<\/mo><\/mrow><\/mrow><\/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\"><mi>x<\/mi><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mrow><mn>\u22123<\/mn><\/mrow><\/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\"><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>9<\/mn><\/mrow><\/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<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/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><mrow><mo>(<\/mo><mrow><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow><mo>)<\/mo><\/mrow><\/mrow><\/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\"><mi>x<\/mi><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mn>3<\/mn><\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mrow><mo>(<\/mo><mrow><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow><mo>)<\/mo><\/mrow><\/mrow><\/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\"><mi>x<\/mi><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mrow><mn>\u22123<\/mn><\/mrow><\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1759726\">The solutions are<\/p>\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>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22123.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22123.<\/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-id2029177\" 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_05_03\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1322224\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2751610\">Solve by factoring:<\/p>\r\n\r\n<math display=\"inline\"><semantics><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>25<\/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>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>25<\/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-id2876101\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Solving a Quadratic Equation by Factoring when the Leading Coefficient is not 1<\/h3>\r\n<p id=\"fs-id2802890\">When the leading coefficient is not 1, we factor a quadratic equation using the method called grouping, which requires four terms. With the equation in standard form, let\u2019s review the grouping procedures:<\/p>\r\n\r\n<ol id=\"fs-id1445659\" type=\"1\">\r\n \t<li>With the quadratic in standard form,\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>multiply\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u22c5<\/mo><mi>c<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>\u22c5<\/mo><mi>c<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li>Find two numbers whose product equals\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mi>c<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mi>c<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and whose sum equals\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>b<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>b<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li>Rewrite the equation replacing the\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>b<\/mi><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>b<\/mi><mi>x<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>term with two terms using the numbers found in step 2 as coefficients of <em data-effect=\"italics\">x.<\/em><\/li>\r\n \t<li>Factor the first two terms and then factor the last two terms. The expressions in parentheses must be exactly the same to use grouping.<\/li>\r\n \t<li>Factor out the expression in parentheses.<\/li>\r\n \t<li>Set the expressions equal to zero and solve for the variable.<\/li>\r\n<\/ol>\r\n<div id=\"Example_02_05_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-id2917470\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2505105\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving a Quadratic Equation Using Grouping<\/h3>\r\n<p id=\"fs-id2638849\">Use grouping to factor and solve the quadratic equation:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>4<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>15<\/mn><mi>x<\/mi><mo>+<\/mo><mn>9<\/mn><mo>=<\/mo><mn>0.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>4<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>15<\/mn><mi>x<\/mi><mo>+<\/mo><mn>9<\/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-id1223661\" 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-id1558604\">First, multiply<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mi>c<\/mi><mo>:<\/mo><mn>4<\/mn><mrow><mo>(<\/mo>\r\n<mn>9<\/mn>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>36.<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mi>c<\/mi><mo>:<\/mo><mn>4<\/mn><mrow><mo>(<\/mo>\r\n<mn>9<\/mn>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>36.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>Then list the factors of\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>36.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>36.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<div id=\"fs-id1449438\" 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<mn>1<\/mn><mo>\u22c5<\/mo><mn>36<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>2<\/mn><mo>\u22c5<\/mo><mn>18<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>3<\/mn><mo>\u22c5<\/mo><mn>12<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>4<\/mn><mo>\u22c5<\/mo><mn>9<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>6<\/mn><mo>\u22c5<\/mo><mn>6<\/mn>\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<mn>1<\/mn><mo>\u22c5<\/mo><mn>36<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>2<\/mn><mo>\u22c5<\/mo><mn>18<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>3<\/mn><mo>\u22c5<\/mo><mn>12<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>4<\/mn><mo>\u22c5<\/mo><mn>9<\/mn>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mn>6<\/mn><mo>\u22c5<\/mo><mn>6<\/mn>\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-id3107108\">The only pair of factors that sums to<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>15<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>15<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mn>12.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><mo>+<\/mo><mn>12.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>Rewrite the equation replacing the <em data-effect=\"italics\">b <\/em>term,\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>15<\/mn><mi>x<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>15<\/mn><mi>x<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>with two terms using 3 and 12 as coefficients of <em data-effect=\"italics\">x<\/em>. Factor the first two terms, and then factor the last two terms.\r\n<div id=\"fs-id1417674\" 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>4<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>12<\/mn><mi>x<\/mi><mo>+<\/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<mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/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><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/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<\/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>4<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>12<\/mn><mi>x<\/mi><mo>+<\/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<mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/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><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/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<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1579301\">Solve using the zero-product property.<\/p>\r\n\r\n<div id=\"fs-id2502349\" 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\"><mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/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><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/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\"><mi>x<\/mi><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/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\"><mi>x<\/mi><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mrow><mo>\u2212<\/mo><mn>3<\/mn><\/mrow><\/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\"><mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/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><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/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\"><mi>x<\/mi><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\"><mrow><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/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\"><mi>x<\/mi><\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd columnalign=\"left\"><mrow><mo>\u2212<\/mo><mn>3<\/mn><\/mrow><\/mtd>\r\n<\/mtr>\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2432037\">The solutions are<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo>\u2212<\/mo><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>\r\n<mo>\u2212<\/mo><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>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22123.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22123.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>See <a class=\"autogenerated-content\" href=\"2-5-quadratic-equations#Figure_02_05_003\">Figure 3<\/a>.\r\n<div id=\"Figure_02_05_003\" class=\"os-figure\">\r\n<figure class=\"small\" data-id=\"Figure_02_05_003\"><span id=\"fs-id1269396\" data-type=\"media\" data-alt=\"Coordinate plane with the x-axis ranging from negative 6 to 2 with every other tick mark labeled and the y-axis ranging from negative 6 to 2 with each tick mark numbered. The equation: four x squared plus fifteen x plus nine is graphed with its x-intercepts: (-3\/4,0) and (-3,0) plotted as well.\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Coordinate plane with the x-axis ranging from negative 6 to 2 with every other tick mark labeled and the y-axis ranging from negative 6 to 2 with each tick mark numbered. The equation: four x squared plus fifteen x plus nine is graphed with its x-intercepts: (-3\/4,0) and (-3,0) plotted as well.\" width=\"487\" height=\"433\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/c8c54dba0d1717875901655bddf937a8b9e3db6a\" \/>\r\n<\/span><\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span>\r\n<span class=\"os-number\">3<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1554615\" 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_05_04\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2629095\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2697369\">Solve using factoring by grouping:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>12<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>11<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>12<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>11<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/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>\r\n<\/section><\/div>\r\n<div id=\"Example_02_05_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-id3115239\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1561758\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving a Polynomial of Higher Degree by Factoring<\/h3>\r\n<p id=\"fs-id3182370\">Solve the equation by factoring:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22123<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>3<\/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>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>=<\/mo><mn>0.<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22123<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>3<\/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>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>=<\/mo><mn>0.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1400268\" 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-id2440018\">This equation does not look like a quadratic, as the highest power is 3, not 2. Recall that the first thing we want to do when solving any equation is to factor out the GCF, if one exists. And it does here. We can factor out<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo>\u2212<\/mo><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo>\u2212<\/mo><mi>x<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>from all of the terms and then proceed with grouping.\r\n<div id=\"fs-id2367474\" 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>\u22123<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>3<\/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>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/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<mo>\u2212<\/mo><mi>x<\/mi><mo>(<\/mo><mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd 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>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>\u22123<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>3<\/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>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/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<mo>\u2212<\/mo><mi>x<\/mi><mo>(<\/mo><mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>)<\/mo><\/mrow>\r\n<\/mtd>\r\n<mtd><mo>=<\/mo><\/mtd>\r\n<mtd 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-id1752856\">Use grouping on the expression in parentheses.<\/p>\r\n\r\n<div id=\"fs-id2869363\" 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>\u2212<\/mo><mi>x<\/mi><mo>(<\/mo><mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/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<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">[<\/mo><mn>3<\/mn><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><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>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/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>&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>\u2212<\/mo><mi>x<\/mi><mo>(<\/mo><mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/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<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mo>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">[<\/mo><mn>3<\/mn><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><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>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/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>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1552054\">Now, we use the zero-product property. Notice that we have three factors.<\/p>\r\n\r\n<div id=\"fs-id1200475\" 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>\u2212<\/mo><mi>x<\/mi><\/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<mn>3<\/mn><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>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<mo>\u2212<\/mo><mfrac>\r\n<mn>2<\/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<mi>x<\/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<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<mo>\u2212<\/mo><mi>x<\/mi><\/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<mn>3<\/mn><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>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<mo>\u2212<\/mo><mfrac>\r\n<mn>2<\/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<mi>x<\/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<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-id2655312\">The solutions are<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>0<\/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<mo>\u2212<\/mo><mfrac>\r\n<mn>2<\/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>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<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-id2500651\" 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_05_05\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1254757\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1254758\">Solve by factoring:<\/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>11<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>10<\/mn><mi>x<\/mi><mo>=<\/mo><mn>0.<\/mn>\r\n<\/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>11<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>10<\/mn><mi>x<\/mi><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>\r\n<\/section><\/div>\r\n<\/section><\/section><section id=\"fs-id1402144\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using the Square Root Property<\/h2>\r\n<p id=\"fs-id1923792\">When there is no linear term in the equation, another method of solving a quadratic equation is by using the <span id=\"term-00006\" data-type=\"term\">square root property<\/span>, in which we isolate the<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>term and take the square root of the number on the other side of the equals sign. Keep in mind that sometimes we may have to manipulate the equation to isolate the\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>term so that the square root property can be used.\r\n<div id=\"fs-id1831215\" 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=\"\">The Square Root Property<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1569584\">With the<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>term isolated, the square root property states that:\r\n<div id=\"fs-id1499182\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mtext>if<\/mtext><mspace width=\"0.5em\"><\/mspace><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mi>k<\/mi><mo>,<\/mo><mtext>then<\/mtext><mspace width=\"0.5em\"><\/mspace><mi>x<\/mi><mo>=<\/mo><mo>\u00b1<\/mo><msqrt>\r\n<mi>k<\/mi>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mtext>if<\/mtext><mspace width=\"0.5em\"><\/mspace><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mi>k<\/mi><mo>,<\/mo><mtext>then<\/mtext><mspace width=\"0.5em\"><\/mspace><mi>x<\/mi><mo>=<\/mo><mo>\u00b1<\/mo><msqrt>\r\n<mi>k<\/mi>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id2875920\">where <em data-effect=\"italics\">k <\/em>is a nonzero real number.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1521248\" 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-id1155370\"><strong>Given a quadratic equation with an <\/strong><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>term but no\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>term, use the square root property to solve it.\r\n<ol id=\"fs-id3081167\" type=\"1\">\r\n \t<li>Isolate the\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>term on one side of the equal sign.<\/li>\r\n \t<li>Take the square root of both sides of the equation, putting a\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo>\u00b1<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo>\u00b1<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>sign before the expression on the side opposite the squared term.<\/li>\r\n \t<li>Simplify the numbers on the side with the\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo>\u00b1<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo>\u00b1<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>sign.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_05_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-id1905142\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1579057\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving a Simple Quadratic Equation Using the Square Root Property<\/h3>\r\n<p id=\"fs-id1467277\">Solve the quadratic using the square root property:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>8.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\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<details id=\"fs-id1514577\" 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-id1515767\">Take the square root of both sides, and then simplify the radical. Remember to use a<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo>\u00b1<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo>\u00b1<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>sign before the radical symbol.\r\n<div id=\"fs-id2437570\" 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<\/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<\/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<mo>\u00b1<\/mo><msqrt>\r\n<mn>8<\/mn>\r\n<\/msqrt>\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\">\r\n<mrow>\r\n<mo>\u00b1<\/mo><mn>2<\/mn><msqrt>\r\n<mn>2<\/mn>\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 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<\/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<\/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<mo>\u00b1<\/mo><msqrt>\r\n<mn>8<\/mn>\r\n<\/msqrt>\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\">\r\n<mrow>\r\n<mo>\u00b1<\/mo><mn>2<\/mn><msqrt>\r\n<mn>2<\/mn>\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<p id=\"fs-id2437670\">The solutions are<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn><msqrt>\r\n<mn>2<\/mn>\r\n<\/msqrt>\r\n<mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn><msqrt>\r\n<mn>2<\/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<mn>\u22122<\/mn><msqrt>\r\n<mn>2<\/mn>\r\n<\/msqrt>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22122<\/mn><msqrt>\r\n<mn>2<\/mn>\r\n<\/msqrt>\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=\"Example_02_05_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-id862941\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1340166\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving a Quadratic Equation Using the Square Root Property<\/h3>\r\n<p id=\"fs-id2497523\">Solve the quadratic equation:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>4<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mtext>7.<\/mtext><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>4<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mtext>7.<\/mtext><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id1482615\" 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-id1568725\">First, isolate the<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>term. Then take the square root of both sides.\r\n<div id=\"fs-id2317298\" 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>4<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<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>7<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>4<\/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>6<\/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<mrow>\r\n<mfrac>\r\n<mn>6<\/mn>\r\n<mn>4<\/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><mfrac>\r\n<mrow>\r\n<msqrt>\r\n<mn>6<\/mn>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\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>4<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<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>7<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<mn>4<\/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>6<\/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<mrow>\r\n<mfrac>\r\n<mn>6<\/mn>\r\n<mn>4<\/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><mfrac>\r\n<mrow>\r\n<msqrt>\r\n<mn>6<\/mn>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\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-id1522278\">The solutions are<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<msqrt>\r\n<mn>6<\/mn>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>, <\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<msqrt>\r\n<mn>6<\/mn>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<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<mrow>\r\n<msqrt>\r\n<mn>6<\/mn>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mn>2<\/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<mrow>\r\n<msqrt>\r\n<mn>6<\/mn>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mn>2<\/mn>\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-id1257773\" 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_05_06\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1913154\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1913155\">Solve the quadratic equation using the square root property:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>3<\/mn><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>15.<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>15.<\/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-id1956704\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Completing the Square<\/h2>\r\n<p id=\"fs-id1793627\">Not all quadratic equations can be factored or can be solved in their original form using the square root property. In these cases, we may use a method for solving a <span id=\"term-00007\" class=\"no-emphasis\" data-type=\"term\">quadratic equation<\/span> known as <span id=\"term-00008\" data-type=\"term\">completing the square<\/span>. Using this method, we add or subtract terms to both sides of the equation until we have a perfect square trinomial on one side of the equal sign. We then apply the square root property. To complete the square, the leading coefficient, <em data-effect=\"italics\">a<\/em>, must equal 1. If it does not, then divide the entire equation by <em data-effect=\"italics\">a<\/em>. Then, we can use the following procedures to solve a quadratic equation by completing the square.<\/p>\r\n<p id=\"fs-id1538204\">We will use the example<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><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>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>to illustrate each step.\r\n<ol id=\"fs-id1234074\" type=\"1\">\r\n \t<li>\r\n<p id=\"fs-id1583461\">Given a quadratic equation that cannot be factored, and with<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>first add or subtract the constant term to the right side of the equal sign.\r\n<div id=\"fs-id1512735\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>=<\/mo><mn>\u22121<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>=<\/mo><mn>\u22121<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div><\/li>\r\n \t<li>\r\n<p id=\"fs-id1175291\">Multiply the <em data-effect=\"italics\">b <\/em>term by<\/p>\r\n\r\n<math display=\"inline\"><semantics><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<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and square it.\r\n<div id=\"fs-id1410870\" 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<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo stretchy=\"false\">(<\/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>2<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mn>2<\/mn>\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>4<\/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<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo stretchy=\"false\">(<\/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>2<\/mn>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\r\n<mtd columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mn>2<\/mn>\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>4<\/mn>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div><\/li>\r\n \t<li>\r\n<p id=\"fs-id1514197\">Add<\/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<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>b<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/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<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mi>b<\/mi>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>to both sides of the equal sign and simplify the right side. We have\r\n<div id=\"fs-id1526634\" 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>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><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<mrow>\r\n<mo>\u2212<\/mo><mn>1<\/mn><mo>+<\/mo><mn>4<\/mn><\/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<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><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>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<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><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<mrow>\r\n<mo>\u2212<\/mo><mn>1<\/mn><mo>+<\/mo><mn>4<\/mn><\/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<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><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>3<\/mn>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div><\/li>\r\n \t<li>\r\n<p id=\"fs-id1538635\">The left side of the equation can now be factored as a perfect square.<\/p>\r\n\r\n<div id=\"fs-id1719027\" 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>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><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>3<\/mn>\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<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/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<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<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><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>3<\/mn>\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<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/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<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><\/li>\r\n \t<li>\r\n<p id=\"fs-id2707455\">Use the square root property and solve.<\/p>\r\n\r\n<div id=\"fs-id2385243\" 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<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/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<mo>\u00b1<\/mo><msqrt>\r\n<mn>3<\/mn>\r\n<\/msqrt>\r\n<\/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<mo>\u00b1<\/mo><msqrt>\r\n<mn>3<\/mn>\r\n<\/msqrt>\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<mn>\u22122<\/mn><mo>\u00b1<\/mo><msqrt>\r\n<mn>3<\/mn>\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<msqrt>\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/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<mo>\u00b1<\/mo><msqrt>\r\n<mn>3<\/mn>\r\n<\/msqrt>\r\n<\/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<mo>\u00b1<\/mo><msqrt>\r\n<mn>3<\/mn>\r\n<\/msqrt>\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<mn>\u22122<\/mn><mo>\u00b1<\/mo><msqrt>\r\n<mn>3<\/mn>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div><\/li>\r\n \t<li>\r\n<p id=\"fs-id3263921\">The solutions are<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>\u22122<\/mn><mo>+<\/mo><msqrt>\r\n<mn>3<\/mn>\r\n<\/msqrt>\r\n<mo>,<\/mo><mo> <\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22122<\/mn><mo>+<\/mo><msqrt>\r\n<mn>3<\/mn>\r\n<\/msqrt>\r\n<mo>,<\/mo><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>\u22122<\/mn><mo>\u2212<\/mo><msqrt>\r\n<mn>3<\/mn>\r\n<\/msqrt>\r\n<mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>\u22122<\/mn><mo>\u2212<\/mo><msqrt>\r\n<mn>3<\/mn>\r\n<\/msqrt>\r\n<mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/li>\r\n<\/ol>\r\n<div id=\"Example_02_05_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-id1884309\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1199291\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving a Quadratic by Completing the Square<\/h3>\r\n<p id=\"fs-id2979951\">Solve the quadratic equation by completing the square:<\/p>\r\n\r\n<math display=\"inline\"><semantics><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>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/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>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/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-id2933237\" 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-id2523221\">First, move the constant term to the right side of the equal sign.<\/p>\r\n\r\n<div id=\"fs-id1560122\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><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>3<\/mn><mi>x<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1701880\">Then, take<\/p>\r\n\r\n<math display=\"inline\"><semantics><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<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>of the <em data-effect=\"italics\">b <\/em>term and square it.\r\n<div id=\"fs-id1700237\" 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<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo stretchy=\"false\">(<\/mo><mn>\u22123<\/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<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/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<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\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<mfrac>\r\n<mn>9<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\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<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo stretchy=\"false\">(<\/mo><mn>\u22123<\/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<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/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<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\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<mfrac>\r\n<mn>9<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\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-id3116426\">Add the result to both sides of the equal sign.<\/p>\r\n\r\n<div id=\"fs-id1919626\" 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>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\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<mn>5<\/mn><mo>+<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\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<\/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>3<\/mn><mi>x<\/mi><mo>+<\/mo><mfrac>\r\n<mn>9<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\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>5<\/mn><mo>+<\/mo><mfrac>\r\n<mn>9<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\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<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\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<mn>5<\/mn><mo>+<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\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<\/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>3<\/mn><mi>x<\/mi><mo>+<\/mo><mfrac>\r\n<mn>9<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\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>5<\/mn><mo>+<\/mo><mfrac>\r\n<mn>9<\/mn>\r\n<mn>4<\/mn>\r\n<\/mfrac>\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-id2495836\">Factor the left side as a perfect square and simplify the right side.<\/p>\r\n\r\n<div id=\"fs-id3176685\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><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><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>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mn>29<\/mn>\r\n<\/mrow>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/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><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>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mn>29<\/mn>\r\n<\/mrow>\r\n<mn>4<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1482511\">Use the square root property and solve.<\/p>\r\n\r\n<div id=\"fs-id1545239\" 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<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/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<mo>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>29<\/mn><\/mrow>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/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<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/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<mrow>\r\n<mo>\u00b1<\/mo><mfrac>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>29<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mn>2<\/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<mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>\u00b1<\/mo><mfrac>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>29<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\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<msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/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<mo>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>29<\/mn><\/mrow>\r\n<mn>4<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<\/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<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/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<mrow>\r\n<mo>\u00b1<\/mo><mfrac>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>29<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mn>2<\/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<mfrac>\r\n<mn>3<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mo>\u00b1<\/mo><mfrac>\r\n<mrow>\r\n<msqrt>\r\n<mrow>\r\n<mn>29<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\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-id2628693\">The solutions are<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo>\r\n<msqrt>\r\n<mn>29<\/mn>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mo>+<\/mo>\r\n<msqrt>\r\n<mn>29<\/mn>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn>\r\n<\/mrow>\r\n<\/mfrac><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\nand\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mo>-<\/mo>\r\n<msqrt>\r\n<mn>29<\/mn>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn>\r\n<\/mrow>\r\n<\/mfrac>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mfrac>\r\n<mrow>\r\n<mn>3<\/mn><mo>-<\/mo>\r\n<msqrt>\r\n<mn>29<\/mn>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn>\r\n<\/mrow>\r\n<\/mfrac><\/annotation-xml><\/semantics><\/math>.\r\n\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2410455\" 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_05_07\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3086043\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3086044\">Solve by completing the square:<\/p>\r\n\r\n<math display=\"inline\"><semantics><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>6<\/mn><mi>x<\/mi><mo>=<\/mo><mn>13.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>6<\/mn><mi>x<\/mi><mo>=<\/mo><mn>13.<\/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-id2979976\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using the Quadratic Formula<\/h2>\r\n<p id=\"fs-id2367499\">The fourth method of solving a <span id=\"term-00009\" class=\"no-emphasis\" data-type=\"term\">quadratic equation<\/span> is by using the <span id=\"term-00010\" class=\"no-emphasis\" data-type=\"term\">quadratic formula<\/span>, a formula that will solve all quadratic equations. Although the quadratic formula works on any quadratic equation in standard form, it is easy to make errors in substituting the values into the formula. Pay close attention when substituting, and use parentheses when inserting a negative number.<\/p>\r\n<p id=\"fs-id1335013\">We can derive the quadratic formula by <span id=\"term-00011\" class=\"no-emphasis\" data-type=\"term\">completing the square<\/span>. We will assume that the leading coefficient is positive; if it is negative, we can multiply the equation by<\/p>\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>and obtain a positive <em data-effect=\"italics\">a<\/em>. Given\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/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<mi>a<\/mi><mo>\u2260<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>\u2260<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>we will complete the square as follows:\r\n<ol id=\"fs-id1894962\" type=\"1\">\r\n \t<li>\r\n<p id=\"fs-id2429906\">First, move the constant term to the right side of the equal sign:<\/p>\r\n\r\n<div id=\"fs-id1547656\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>c<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>c<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div><\/li>\r\n \t<li>\r\n<p id=\"fs-id1569987\">As we want the leading coefficient to equal 1, divide through by <em data-effect=\"italics\">a<\/em>:<\/p>\r\n\r\n<div id=\"fs-id1528274\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mfrac>\r\n<mi>b<\/mi>\r\n<mi>a<\/mi>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mi>c<\/mi>\r\n<mi>a<\/mi>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mfrac>\r\n<mi>b<\/mi>\r\n<mi>a<\/mi>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\r\n<mi>c<\/mi>\r\n<mi>a<\/mi>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div><\/li>\r\n \t<li>\r\n<p id=\"fs-id2906392\">Then, find<\/p>\r\n\r\n<math display=\"inline\"><semantics><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<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>of the middle term, and add\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<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mfrac>\r\n<mi>b<\/mi>\r\n<mi>a<\/mi>\r\n<\/mfrac>&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><mfrac>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<mrow>\r\n<mn>4<\/mn><msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/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<mfrac>\r\n<mn>1<\/mn>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<mfrac>\r\n<mi>b<\/mi>\r\n<mi>a<\/mi>\r\n<\/mfrac>&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><mfrac>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<mrow>\r\n<mn>4<\/mn><msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>to both sides of the equal sign:\r\n<div id=\"fs-id2443764\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mfrac>\r\n<mi>b<\/mi>\r\n<mi>a<\/mi>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>+<\/mo><mfrac>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<mrow>\r\n<mn>4<\/mn><msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<mrow>\r\n<mn>4<\/mn><msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mi>c<\/mi>\r\n<mi>a<\/mi>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mfrac>\r\n<mi>b<\/mi>\r\n<mi>a<\/mi>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>+<\/mo><mfrac>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<mrow>\r\n<mn>4<\/mn><msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mfrac>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<mrow>\r\n<mn>4<\/mn><msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mi>c<\/mi>\r\n<mi>a<\/mi>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div><\/li>\r\n \t<li>\r\n<p id=\"fs-id1719390\">Next, write the left side as a perfect square. Find the common denominator of the right side and write it as a single fraction:<\/p>\r\n\r\n<div id=\"fs-id3148450\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><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><mfrac>\r\n<mi>b<\/mi>\r\n<mrow>\r\n<mn>2<\/mn><mi>a<\/mi>\r\n<\/mrow>\r\n<\/mfrac>&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><mfrac>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mn>4<\/mn><msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/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><mfrac>\r\n<mi>b<\/mi>\r\n<mrow>\r\n<mn>2<\/mn><mi>a<\/mi>\r\n<\/mrow>\r\n<\/mfrac>&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><mfrac>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi>\r\n<\/mrow>\r\n<mrow>\r\n<mn>4<\/mn><msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div><\/li>\r\n \t<li>\r\n<p id=\"fs-id2307686\">Now, use the square root property, which gives<\/p>\r\n\r\n<div id=\"fs-id1937587\" 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>x<\/mi><mo>+<\/mo><mfrac>\r\n<mi>b<\/mi>\r\n<mrow>\r\n<mn>2<\/mn><mi>a<\/mi><\/mrow>\r\n<\/mfrac>\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>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow>\r\n<mrow>\r\n<mn>4<\/mn><msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msqrt>\r\n<\/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><mfrac>\r\n<mi>b<\/mi>\r\n<mrow>\r\n<mn>2<\/mn><mi>a<\/mi><\/mrow>\r\n<\/mfrac>\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<mfrac>\r\n<mrow>\r\n<mo>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mi>a<\/mi><\/mrow>\r\n<\/mfrac>\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<mi>x<\/mi><mo>+<\/mo><mfrac>\r\n<mi>b<\/mi>\r\n<mrow>\r\n<mn>2<\/mn><mi>a<\/mi><\/mrow>\r\n<\/mfrac>\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>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow>\r\n<mrow>\r\n<mn>4<\/mn><msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/msqrt>\r\n<\/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><mfrac>\r\n<mi>b<\/mi>\r\n<mrow>\r\n<mn>2<\/mn><mi>a<\/mi><\/mrow>\r\n<\/mfrac>\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<mfrac>\r\n<mrow>\r\n<mo>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mi>a<\/mi><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div><\/li>\r\n \t<li>\r\n<p id=\"fs-id1977996\">Finally, add<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mi>b<\/mi>\r\n<mrow>\r\n<mn>2<\/mn><mi>a<\/mi>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mi>b<\/mi>\r\n<mrow>\r\n<mn>2<\/mn><mi>a<\/mi>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>to both sides of the equation and combine the terms on the right side. Thus,\r\n<div id=\"fs-id2440094\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mi>a<\/mi>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mi>a<\/mi>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div><\/li>\r\n<\/ol>\r\n<div id=\"fs-id2500151\" 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=\"\">The Quadratic Formula<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1278348\">Written in standard form,<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>any quadratic equation can be solved using the <span id=\"term-00012\" data-type=\"term\">quadratic formula<\/span>:\r\n<div id=\"fs-id1959865\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mi>a<\/mi>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi><mo>=<\/mo><mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi>\r\n<\/mrow>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mi>a<\/mi>\r\n<\/mrow>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1930451\">where <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are real numbers and<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>\u2260<\/mo><mn>0.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>\u2260<\/mo><mn>0.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1480868\" 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-id2385664\"><strong>Given a quadratic equation, solve it using the quadratic formula<\/strong><\/p>\r\n\r\n<ol id=\"fs-id1973820\" type=\"1\">\r\n \t<li>Make sure the equation is in standard form:\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li>Make note of the values of the coefficients and constant term,\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>,<\/mo><mi>b<\/mi><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>,<\/mo><mi>b<\/mi><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>c<\/mi><mo>.<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>c<\/mi><mo>.<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\r\n \t<li>Carefully substitute the values noted in step 2 into the equation. To avoid needless errors, use parentheses around each number input into the formula.<\/li>\r\n \t<li>Calculate and solve.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_05_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-id2979701\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2979703\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solve the Quadratic Equation Using the Quadratic Formula<\/h3>\r\n<p id=\"fs-id2442582\">Solve the quadratic equation:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><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>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><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<details id=\"fs-id3034009\" 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-id1800418\">Identify the coefficients:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo><mi>b<\/mi><mo>=<\/mo><mn>5<\/mn><mo>,<\/mo><mi>c<\/mi><mo>=<\/mo><mn>1.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo><mi>b<\/mi><mo>=<\/mo><mn>5<\/mn><mo>,<\/mo><mi>c<\/mi><mo>=<\/mo><mn>1.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>Then use the quadratic formula.\r\n<div id=\"fs-id1813797\" 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<mi>x<\/mi>\r\n<\/mtd>\r\n<mtd rowalign=\"center\">\r\n<mo>=<\/mo>\r\n<\/mtd>\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mfrac>\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\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>5<\/mn><mo>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<mn>25<\/mn><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\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<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>5<\/mn><mo>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<mn>21<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\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 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=\"right\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/mfrac>\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\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>5<\/mn><mo>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<mn>25<\/mn><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\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<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>5<\/mn><mo>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<mn>21<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/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=\"Example_02_05_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-id1561272\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1457087\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Solving a Quadratic Equation with the Quadratic Formula<\/h3>\r\n<p id=\"fs-id1457092\">Use the quadratic formula to solve<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/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-id1717549\" 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-id1717551\">First, we identify the coefficients:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo><mi>b<\/mi><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo><mi>b<\/mi><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>c<\/mi><mo>=<\/mo><mn>2.<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>c<\/mi><mo>=<\/mo><mn>2.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<p id=\"fs-id1531480\">Substitute these values into the quadratic formula.<\/p>\r\n\r\n<div id=\"fs-id2486075\" 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<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<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mi>a<\/mi><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\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<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mfrac>\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\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>1<\/mn><mo>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<mn>1<\/mn><mo>\u2212<\/mo><mn>8<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\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\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>1<\/mn><mo>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>7<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\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\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>1<\/mn><mo>\u00b1<\/mo><mi>i<\/mi><msqrt>\r\n<mn>7<\/mn>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\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 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<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mi>a<\/mi><\/mrow>\r\n<\/mfrac>\r\n<\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr>\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<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mrow>\r\n<mn>2<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mfrac>\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\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>1<\/mn><mo>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<mn>1<\/mn><mo>\u2212<\/mo><mn>8<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\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\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>1<\/mn><mo>\u00b1<\/mo><msqrt>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>7<\/mn><\/mrow>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\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\">\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>1<\/mn><mo>\u00b1<\/mo><mi>i<\/mi><msqrt>\r\n<mn>7<\/mn>\r\n<\/msqrt>\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>\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-id1798703\">The solutions to the equation are<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>1<\/mn><mo>+<\/mo><mi>i<\/mi><msqrt>\r\n<mn>7<\/mn>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>1<\/mn><mo>+<\/mo><mi>i<\/mi><msqrt>\r\n<mn>7<\/mn>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\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<mrow>\r\n<mo>\u2212<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>i<\/mi><msqrt>\r\n<mn>7<\/mn>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mrow>\r\n<mo>\u2212<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>i<\/mi><msqrt>\r\n<mn>7<\/mn>\r\n<\/msqrt>&nbsp;\r\n\r\n<\/mrow>\r\n<mn>2<\/mn>\r\n<\/mfrac>&nbsp;\r\n\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-id2370081\" 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_05_08\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2933152\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2933153\">Solve the quadratic equation using the quadratic formula:<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>9<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>3<\/mn><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<mn>9<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>3<\/mn><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>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2932184\" data-depth=\"1\">\r\n<h2 data-type=\"title\">The Discriminant<\/h2>\r\n<p id=\"fs-id1965271\">The <span id=\"term-00013\" class=\"no-emphasis\" data-type=\"term\">quadratic formula<\/span> not only generates the solutions to a quadratic equation, it tells us about the nature of the solutions when we consider the <span id=\"term-00014\" class=\"no-emphasis\" data-type=\"term\">discriminant<\/span>, or the expression under the radical,<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>.<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>The discriminant tells us whether the solutions are real numbers or complex numbers, and how many solutions of each type to expect. <a class=\"autogenerated-content\" href=\"2-5-quadratic-equations#Table_02_05_01\">Table 1<\/a> relates the value of the discriminant to the solutions of a quadratic equation.\r\n<div id=\"Table_02_05_01\" class=\"os-table\">\r\n<table data-id=\"Table_02_05_01\">\r\n<thead>\r\n<tr>\r\n<th scope=\"col\" data-align=\"center\">Value of Discriminant<\/th>\r\n<th scope=\"col\" data-align=\"center\">Results<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<td data-align=\"center\">One rational solution (double solution)<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>perfect square<\/td>\r\n<td data-align=\"center\">Two rational solutions<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>not a perfect square<\/td>\r\n<td data-align=\"center\">Two irrational solutions<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>&lt;<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>&lt;<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\r\n<td data-align=\"center\">Two complex solutions<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span>\r\n<span class=\"os-number\">1<\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id2495380\" 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=\"\">The Discriminant<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"os-note-body\">\r\n<p id=\"fs-id1150951\">For<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>, where\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>,\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>b<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>b<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>, and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>c<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>c<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>are real numbers, the <span id=\"term-00015\" data-type=\"term\">discriminant<\/span> is the expression under the radical in the quadratic formula:\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>.<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>It tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"Example_02_05_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-id1340700\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1340702\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Using the Discriminant to Find the Nature of the Solutions to a Quadratic Equation<\/h3>\r\n<p id=\"fs-id2515703\">Use the discriminant to find the nature of the solutions to the following quadratic equations:<\/p>\r\n\r\n<ol id=\"fs-id2515706\" 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<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn>\r\n<\/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<mn>8<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>14<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>8<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>14<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>=<\/mo><mn>0<\/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<mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>5<\/mn><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<mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/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<mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>10<\/mn><mi>x<\/mi><mo>+<\/mo><mn>15<\/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>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>10<\/mn><mi>x<\/mi><mo>+<\/mo><mn>15<\/mn><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-id2508928\" 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-id2508930\">Calculate the discriminant<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><\/annotation-xml><\/semantics><\/math>for each equation and state the expected type of solutions.\r\n<ol id=\"fs-id1385650\" class=\"circled\" type=\"1\">\r\n \t<li><span class=\"token\">\u24d0<\/span>\r\n<p id=\"fs-id1699199\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<p id=\"fs-id2390532\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>=<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mn>4<\/mn>\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<mn>1<\/mn>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mn>4<\/mn>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>0.<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>=<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mn>4<\/mn>\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<mn>1<\/mn>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mn>4<\/mn>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>0.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>There will be one rational double solution.<\/li>\r\n \t<li><span class=\"token\">\u24d1<\/span>\r\n<p id=\"fs-id2933880\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>8<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>14<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>8<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>14<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<p id=\"fs-id2422396\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>=<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>14<\/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<mn>8<\/mn>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mn>3<\/mn>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>100.<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>=<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>14<\/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<mn>8<\/mn>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mn>3<\/mn>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>100.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>As\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>100<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>100<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is a perfect square, there will be two rational solutions.<\/li>\r\n \t<li><span class=\"token\">\u24d2<\/span>\r\n<p id=\"fs-id1690680\"><\/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>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>5<\/mn><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<mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<p id=\"fs-id1804761\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>=<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22125<\/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<mn>3<\/mn>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22122<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>49.<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>=<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22125<\/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<mn>3<\/mn>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u22122<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>49.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>As\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>49<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>49<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is a perfect square, there will be two rational solutions.<\/li>\r\n \t<li><span class=\"token\">\u24d3<\/span>\r\n<p id=\"fs-id2016618\"><\/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>2<\/mn>\r\n<\/msup>\r\n<mn>\u221210<\/mn><mi>x<\/mi><mo>+<\/mo><mn>15<\/mn><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>3<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mn>\u221210<\/mn><mi>x<\/mi><mo>+<\/mo><mn>15<\/mn><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<p id=\"fs-id2905584\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>=<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u221210<\/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<mn>3<\/mn>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>15<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>\u221280.<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>=<\/mo><msup>\r\n<mrow>\r\n<mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>\u221210<\/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<mn>3<\/mn>\r\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\r\n<mrow>\r\n<mn>15<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>\u221280.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>There will be two complex solutions.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2443208\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using the Pythagorean Theorem<\/h2>\r\n<p id=\"fs-id2454436\">One of the most famous formulas in mathematics is the <span id=\"term-00016\" data-type=\"term\">Pythagorean Theorem<\/span>. It is based on a right triangle, and states the relationship among the lengths of the sides as<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>c<\/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>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>c<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>where\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>b<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>b<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>refer to the legs of a right triangle adjacent to the\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>90\u00b0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>90\u00b0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>angle, and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>c<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>c<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>refers to the hypotenuse. It has immeasurable uses in architecture, engineering, the sciences, geometry, trigonometry, and algebra, and in everyday applications.\r\n<p id=\"fs-id2797112\">We use the Pythagorean Theorem to solve for the length of one side of a triangle when we have the lengths of the other two. Because each of the terms is squared in the theorem, when we are solving for a side of a triangle, we have a quadratic equation. We can use the methods for solving quadratic equations that we learned in this section to solve for the missing side.<\/p>\r\n<p id=\"fs-id2382116\">The Pythagorean Theorem is given as<\/p>\r\n\r\n<div id=\"fs-id2797115\" 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<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>c<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><msup>\r\n<mi>c<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\r\n<p id=\"fs-id1297273\">where<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>b<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>b<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>refer to the legs of a right triangle adjacent to the\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mrow>\r\n<mn>90<\/mn>\r\n<\/mrow>\r\n<mo>\u2218<\/mo>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mrow>\r\n<mn>90<\/mn>\r\n<\/mrow>\r\n<mo>\u2218<\/mo>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>angle, and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>c<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>c<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>refers to the hypotenuse, as shown in <a class=\"autogenerated-content\" href=\"2-5-quadratic-equations#Figure_02_05_004\">Figure 4<\/a>.\r\n<div id=\"Figure_02_05_004\" class=\"os-figure\">\r\n<figure class=\"small\" data-id=\"Figure_02_05_004\"><span id=\"fs-id3040390\" data-type=\"media\" data-alt=\"Right triangle with the base labeled: a, the height labeled: b, and the hypotenuse labeled: c\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Right triangle with the base labeled: a, the height labeled: b, and the hypotenuse labeled: c\" width=\"248\" height=\"258\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/118170e707bfd06f5fedcf2dc304080f9308e31b\" \/>\r\n<\/span><\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span>\r\n<span class=\"os-number\">4<\/span><\/div>\r\n<\/div>\r\n<div id=\"Example_02_05_12\" 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\">12<\/span><\/h2>\r\n<\/header><section>\r\n<div class=\"body\">\r\n<div id=\"fs-id2700450\" class=\"unnumbered\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2700452\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<h3 data-type=\"title\">Finding the Length of the Missing Side of a Right Triangle<\/h3>\r\n<p id=\"fs-id1887267\">Find the length of the missing side of the right triangle in <a class=\"autogenerated-content\" href=\"2-5-quadratic-equations#Figure_02_05_005\">Figure 5<\/a>.<\/p>\r\n\r\n<div id=\"Figure_02_05_005\" class=\"os-figure\">\r\n<figure class=\"small\" data-id=\"Figure_02_05_005\"><span id=\"fs-id3155491\" data-type=\"media\" data-alt=\"Right triangle with the base labeled: a, the height labeled: 4, and the hypotenuse labeled 12.\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Right triangle with the base labeled: a, the height labeled: 4, and the hypotenuse labeled 12.\" width=\"320\" height=\"183\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/aa457e668bedf9992e721efdd4bb275c8cc62b1c\" \/>\r\n<\/span><\/figure>\r\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span>\r\n<span class=\"os-number\">5<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<details id=\"fs-id2308574\" 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-id2528218\">As we have measurements for side <em data-effect=\"italics\">b<\/em> and the hypotenuse, the missing side is <em data-effect=\"italics\">a.<\/em><\/p>\r\n\r\n<div id=\"fs-id2294487\" 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>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\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<msup>\r\n<mi>c<\/mi>\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\" columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mn>2<\/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<mo stretchy=\"false\">(<\/mo><mn>12<\/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\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>16<\/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>144<\/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<msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/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<mn>128<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mi>a<\/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<msqrt>\r\n<mrow>\r\n<mn>128<\/mn><\/mrow>\r\n<\/msqrt>\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\">\r\n<mrow>\r\n<mn>8<\/mn><msqrt>\r\n<mn>2<\/mn>\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 rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\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<msup>\r\n<mi>c<\/mi>\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\" columnalign=\"left\">\r\n<mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><msup>\r\n<mrow>\r\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\r\n<mn>2<\/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<mo stretchy=\"false\">(<\/mo><mn>12<\/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\" columnalign=\"right\">\r\n<mrow>\r\n<msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>16<\/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>144<\/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<msup>\r\n<mi>a<\/mi>\r\n<mn>2<\/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<mn>128<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr rowalign=\"center\">\r\n<mtd rowalign=\"center\" columnalign=\"right\">\r\n<mi>a<\/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<msqrt>\r\n<mrow>\r\n<mn>128<\/mn><\/mrow>\r\n<\/msqrt>\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\">\r\n<mrow>\r\n<mn>8<\/mn><msqrt>\r\n<mn>2<\/mn>\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>\r\n<\/section><\/details><\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1918460\" 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_05_09\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2381994\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2381996\">Use the Pythagorean Theorem to solve the right triangle problem: Leg <em data-effect=\"italics\">a <\/em>measures 4 units, leg <em data-effect=\"italics\">b <\/em>measures 3 units. Find the length of the hypotenuse.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1762359\" 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-id2666327\">Access these online resources for additional instruction and practice with quadratic equations.<\/p>\r\n\r\n<ul id=\"fs-id2666330\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/quadreqfactor\" target=\"_blank\" rel=\"noopener nofollow\">Solving Quadratic Equations by Factoring<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/zeroprodprop\" target=\"_blank\" rel=\"noopener nofollow\">The Zero-Product Property<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/complthesqr\" target=\"_blank\" rel=\"noopener nofollow\">Completing the Square<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/quadrformrat\" target=\"_blank\" rel=\"noopener nofollow\">Quadratic Formula with Two Rational Solutions<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/leglengthtri\" target=\"_blank\" rel=\"noopener nofollow\">Length of a leg of a right triangle<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/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\">2.5 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id1685751\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id1685757\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id1752696\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1752698\" 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-id1752696-solution\">1<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1752699\">How do we recognize when an equation is quadratic?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1686335\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1686336\" 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-id1686337\">When we solve a quadratic equation, how many solutions should we always start out seeking? Explain why when solving a quadratic equation in the form<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>we may graph the equation\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mi>a<\/mi><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi><mo>=<\/mo><mi>a<\/mi><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and have no zeroes (<em data-effect=\"italics\">x<\/em>-intercepts).\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2440111\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3094972\" 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-id2440111-solution\">3<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3094973\">When we solve a quadratic equation by factoring, why do we move all terms to one side, having zero on the other side?<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1540842\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1540843\" 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-id1540844\">In the quadratic formula, what is the name of the expression under the radical sign<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>b<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and how does it determine the number of and nature of our solutions?\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2439837\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2439838\" 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-id2439837-solution\">5<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2439839\">Describe two scenarios where using the square root property to solve a quadratic equation would be the most efficient method.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id3039271\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<p id=\"fs-id3070461\">For the following exercises, solve the quadratic equation by factoring.<\/p>\r\n\r\n<div id=\"fs-id3070464\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3070466\" 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-id3070467\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>21<\/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>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><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 id=\"fs-id1227822\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1227823\" 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-id1227822-solution\">7<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1227824\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><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>9<\/mn><mi>x<\/mi><mo>+<\/mo><mn>18<\/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>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>9<\/mn><mi>x<\/mi><mo>+<\/mo><mn>18<\/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-id2029003\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2029004\" 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-id2029005\"><\/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>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>9<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><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<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>9<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/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-id1333066\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1333067\" 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-id1333066-solution\">9<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1333068\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>6<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>17<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>6<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>17<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/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-id2957156\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2957157\" 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-id2957158\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>4<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>12<\/mn><mi>x<\/mi><mo>+<\/mo><mn>8<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>4<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>12<\/mn><mi>x<\/mi><mo>+<\/mo><mn>8<\/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-id2708935\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2708936\" 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-id2708935-solution\">11<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2708937\"><\/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>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>75<\/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>2<\/mn>\r\n<\/msup>\r\n<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-id2933002\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2933003\" 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-id2933004\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>8<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>6<\/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>\r\n<mn>8<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>6<\/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<\/section><\/div>\r\n<div id=\"fs-id1417836\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1417837\" 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-id1417836-solution\">13<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1417838\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>4<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>9<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>4<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>9<\/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-id1762775\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1762776\" 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-id1762777\"><\/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>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>14<\/mn><mi>x<\/mi><mo>=<\/mo><mn>36<\/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>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>14<\/mn><mi>x<\/mi><mo>=<\/mo><mn>36<\/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-id3207565\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3207566\" 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-id3207565-solution\">15<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3207567\"><\/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>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>30<\/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>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>30<\/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-id1929832\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1929833\" 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-id1395974\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>4<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>5<\/mn><mi>x<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>4<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>5<\/mn><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-id1520388\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1520389\" 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-id1520388-solution\">17<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1520390\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>7<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>7<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><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-id2438766\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2438767\" 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-id2438768\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mfrac>\r\n<mi>x<\/mi>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>9<\/mn>\r\n<mi>x<\/mi>\r\n<\/mfrac>\r\n<mo>=<\/mo><mn>2<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mfrac>\r\n<mi>x<\/mi>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>9<\/mn>\r\n<mi>x<\/mi>\r\n<\/mfrac>\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<p id=\"fs-id2370838\">For the following exercises, solve the quadratic equation by using the square root property.<\/p>\r\n\r\n<div id=\"fs-id2370842\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2370843\" 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-id2370842-solution\">19<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1353332\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>36<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>36<\/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-id2049646\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2049648\" 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-id2049649\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>49<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>49<\/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-id1354953\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1354954\" 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-id1354953-solution\">21<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1846535\"><\/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<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>25<\/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<mn>2<\/mn>\r\n<\/msup>\r\n<mo>=<\/mo><mn>25<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2519565\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2519566\" 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-id2519567\"><\/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>=<\/mo><mn>7<\/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>=<\/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-id2977340\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2977341\" 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-id2977340-solution\">23<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2977342\"><\/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<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/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>9<\/mn>\r\n<\/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<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/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>9<\/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-id2643328\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2643329\" 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-id2643330\"><\/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>5<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>2<\/mn>\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>\u2212<\/mo><mn>5<\/mn><\/mrow>\r\n<mo>)<\/mo><\/mrow><\/mrow>\r\n<mn>2<\/mn>\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<p id=\"fs-id2432296\">For the following exercises, solve the quadratic equation by completing the square. Show each step.<\/p>\r\n\r\n<div id=\"fs-id2432300\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2432301\" 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-id2432300-solution\">25<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2432302\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><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>9<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>22<\/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>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>9<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>22<\/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-id3215750\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3215751\" 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-id3215752\"><\/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>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>8<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><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<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>8<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/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-id2980329\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2980330\" 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-id2980329-solution\">27<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2980331\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><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>6<\/mn><mi>x<\/mi><mo>=<\/mo><mn>13<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>6<\/mn><mi>x<\/mi><mo>=<\/mo><mn>13<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2020955\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2020956\" 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-id2020957\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mfrac>\r\n<mn>2<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\r\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mn>3<\/mn>\r\n<\/mfrac>\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-id1840550\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1840551\" 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-id1840550-solution\">29<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1840552\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mi>z<\/mi><mo>=<\/mo><mn>6<\/mn><msup>\r\n<mi>z<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>2<\/mn><mo>+<\/mo><mi>z<\/mi><mo>=<\/mo><mn>6<\/mn><msup>\r\n<mi>z<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\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-id1552019\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1552020\" 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-id1552022\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>6<\/mn><msup>\r\n<mi>p<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>7<\/mn><mi>p<\/mi><mo>\u2212<\/mo><mn>20<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>6<\/mn><msup>\r\n<mi>p<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>7<\/mn><mi>p<\/mi><mo>\u2212<\/mo><mn>20<\/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-id1225471\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1225472\" 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-id1225471-solution\">31<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1225473\"><\/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>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/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>2<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/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<p id=\"fs-id1351674\">For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.<\/p>\r\n\r\n<div id=\"fs-id1351679\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1351680\" 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-id1351681\"><\/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>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><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<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/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-id2762702\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2762703\" 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-id2762702-solution\">33<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2762704\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/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-id2734018\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2734019\" 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-id2734020\"><\/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>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>8<\/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>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>8<\/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-id3142966\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3142967\" 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-id3142966-solution\">35<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3142968\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>9<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>30<\/mn><mi>x<\/mi><mo>+<\/mo><mn>25<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>9<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>30<\/mn><mi>x<\/mi><mo>+<\/mo><mn>25<\/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-id1357817\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1845325\" 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-id1845326\"><\/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>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><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<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/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-id1873764\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1873765\" 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-id1873764-solution\">37<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1873766\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>6<\/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<mn>6<\/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<p id=\"fs-id2947602\">For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state <em data-effect=\"italics\">No Real Solution<\/em>.<\/p>\r\n\r\n<div id=\"fs-id1883375\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1883376\" 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-id1883378\"><\/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>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><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>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/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-id3070364\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3070365\" 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-id3070364-solution\">39<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3070366\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>=<\/mo><mn>4<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>x<\/mi><mo>=<\/mo><mn>4<\/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-id1846559\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1846560\" 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-id1846561\"><\/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>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>8<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><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<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>8<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/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-id1716230\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1716231\" 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-id1716230-solution\">41<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3274906\"><\/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>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/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>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/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-id1873745\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1873746\" 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-id1873747\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/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-id2919049\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2919050\" 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-id2919049-solution\">43<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2919051\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>4<\/mn><mo>+<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mi>x<\/mi>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>\r\n<mo>=<\/mo><mn>0<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>4<\/mn><mo>+<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mi>x<\/mi>\r\n<\/mfrac>\r\n<mo>\u2212<\/mo><mfrac>\r\n<mn>1<\/mn>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mfrac>\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<\/section><section id=\"fs-id1421999\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Technology<\/h3>\r\n<p id=\"fs-id1846578\">For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the <em data-effect=\"italics\">x<\/em>-intercepts) by using <strong>2<sup>nd<\/sup> CALC 2:zero<\/strong>. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth.<\/p>\r\n\r\n<div id=\"fs-id1846592\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1846593\" 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-id1846594\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msub>\r\n<mtext>Y<\/mtext>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>=<\/mo><mn>4<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msub>\r\n<mtext>Y<\/mtext>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>=<\/mo><mn>4<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1204326\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1204327\" 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-id1204326-solution\">45<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1204328\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msub>\r\n<mtext>Y<\/mtext>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>=<\/mo><mn>\u22123<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>8<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msub>\r\n<mtext>Y<\/mtext>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>=<\/mo><mn>\u22123<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>8<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2506957\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2506958\" 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-id2506959\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msub>\r\n<mtext>Y<\/mtext>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>=<\/mo><mn>0.5<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\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<msub>\r\n<mtext>Y<\/mtext>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>=<\/mo><mn>0.5<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\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-id1442695\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1442697\" 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-id1442695-solution\">47<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1442698\">To solve the quadratic equation<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><mo>=<\/mo><mn>4<\/mn><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><mo>=<\/mo><mn>4<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>we can graph these two equations\r\n<p id=\"fs-id2442829\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msub>\r\n<mtext>Y<\/mtext>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>=<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msub>\r\n<mtext>Y<\/mtext>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<mo>=<\/mo><mn>4<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/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 columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msub>\r\n<mtext>Y<\/mtext>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>=<\/mo><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msub>\r\n<mtext>Y<\/mtext>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<mo>=<\/mo><mn>4<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<p id=\"fs-id1240069\">and find the points of intersection. Recall 2<sup>nd<\/sup> CALC 5:intersection. Do this and find the solutions to the nearest tenth.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2697512\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2697513\" 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-id2495814\">To solve the quadratic equation<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>0.3<\/mn><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>4<\/mn><mo>=<\/mo><mn>2<\/mn><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>0.3<\/mn><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>4<\/mn><mo>=<\/mo><mn>2<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>we can graph these two equations\r\n<p id=\"fs-id2413227\"><\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msub>\r\n<mtext>Y<\/mtext>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>=<\/mo><mn>0.3<\/mn><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>4<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msub>\r\n<mtext>Y<\/mtext>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<mo>=<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/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 columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<mtable columnalign=\"left\">\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msub>\r\n<mtext>Y<\/mtext>\r\n<mn>1<\/mn>\r\n<\/msub>\r\n<mo>=<\/mo><mn>0.3<\/mn><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>4<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>\r\n<mtr columnalign=\"left\">\r\n<mtd columnalign=\"left\">\r\n<mrow>\r\n<msub>\r\n<mtext>Y<\/mtext>\r\n<mn>2<\/mn>\r\n<\/msub>\r\n<mo>=<\/mo><mn>2<\/mn><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow>\r\n<\/mtd>\r\n<\/mtr>&nbsp;\r\n\r\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;\r\n<p id=\"fs-id2906629\">and find the points of intersection. Recall 2<sup>nd<\/sup> CALC 5:intersection. Do this and find the solutions to the nearest tenth.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id2498853\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Extensions<\/h3>\r\n<div id=\"fs-id2498858\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2498859\" 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-id2498858-solution\">49<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2498860\">Beginning with the general form of a quadratic equation,<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>a<\/mi><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>a<\/mi><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>solve for <em data-effect=\"italics\">x<\/em> by using the completing the square method, thus deriving the quadratic formula.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2428496\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2428497\" data-type=\"problem\"><span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2428498\">Show that the sum of the two solutions to the quadratic equation is<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow><mo>\u2212<\/mo>\r\n<mfrac>\r\n<mrow>\r\n<mi>b<\/mi>\r\n<\/mrow>\r\n<mi>a<\/mi>\r\n<\/mfrac>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>\u2212<\/mo><mfrac>\r\n<mrow>\r\n<mi>b<\/mi>\r\n<\/mrow>\r\n<mi>a<\/mi>\r\n<\/mfrac><\/annotation-xml><\/semantics><\/math>.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2026640\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2021216\" 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-id2026640-solution\">51<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2021217\">A person has a garden that has a length 10 feet longer than the width. Set up a quadratic equation to find the dimensions of the garden if its area is 119 ft.<sup>2<\/sup>. Solve the quadratic equation to find the length and width.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2644744\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2644745\" data-type=\"problem\"><span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2720880\">Abercrombie and Fitch stock had a price given as<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>P<\/mi><mo>=<\/mo><mn>0.2<\/mn><msup>\r\n<mi>t<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>5.6<\/mn><mi>t<\/mi><mo>+<\/mo><mn>50.2<\/mn><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>P<\/mi><mo>=<\/mo><mn>0.2<\/mn><msup>\r\n<mi>t<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>5.6<\/mn><mi>t<\/mi><mo>+<\/mo><mn>50.2<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>where\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>t<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>t<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is the time in months from 1999 to 2001. (\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>t<\/mi><mo>=<\/mo><mn>1<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>t<\/mi><mo>=<\/mo><mn>1<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is January 1999). Find the two months in which the price of the stock was $30.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id3113026\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id3113027\" 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-id3113026-solution\">53<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id3113028\">Suppose that an equation is given<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>p<\/mi><mo>=<\/mo><mn>\u22122<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>280<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1000<\/mn><mo>,<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>p<\/mi><mo>=<\/mo><mn>\u22122<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>280<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1000<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>where\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>represents the number of items sold at an auction and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>p<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>p<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is the profit made by the business that ran the auction. How many items sold would make this profit a maximum? Solve this by graphing the expression in your graphing utility and finding the maximum using 2<sup>nd<\/sup> CALC maximum. To obtain a good window for the curve, set\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>x<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>x<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>[0,200] and\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>y<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>y<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>[0,10000].\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1894925\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Real-World Applications<\/h3>\r\n<div id=\"fs-id1894930\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2644764\" data-type=\"problem\"><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2644765\">A formula for the normal systolic blood pressure for a man age<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>A<\/mi><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>A<\/mi><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>measured in mmHg, is given as\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>P<\/mi><mo>=<\/mo><mn>0.006<\/mn><msup>\r\n<mi>A<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>0.02<\/mn><mi>A<\/mi><mo>+<\/mo><mn>120.<\/mn><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>P<\/mi><mo>=<\/mo><mn>0.006<\/mn><msup>\r\n<mi>A<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>\u2212<\/mo><mn>0.02<\/mn><mi>A<\/mi><mo>+<\/mo><mn>120.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>Find the age to the nearest year of a man whose normal blood pressure measures 125 mmHg.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1700644\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1700645\" 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-id1700644-solution\">55<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id1700646\">The cost function for a certain company is<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>C<\/mi><mo>=<\/mo><mn>60<\/mn><mi>x<\/mi><mo>+<\/mo><mn>300<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>C<\/mi><mo>=<\/mo><mn>60<\/mn><mi>x<\/mi><mo>+<\/mo><mn>300<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>and the revenue is given by\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>R<\/mi><mo>=<\/mo><mn>100<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>0.5<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>.<\/mo><\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>R<\/mi><mo>=<\/mo><mn>100<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>0.5<\/mn><msup>\r\n<mi>x<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>Recall that profit is revenue minus cost. Set up a quadratic equation and find two values of <em data-effect=\"italics\">x<\/em> (production level) that will create a profit of $300.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2454082\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2454083\" data-type=\"problem\"><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2454084\">A falling object travels a distance given by the formula<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>d<\/mi><mo>=<\/mo><mn>5<\/mn><mi>t<\/mi><mo>+<\/mo><mn>16<\/mn><msup>\r\n<mi>t<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>d<\/mi><mo>=<\/mo><mn>5<\/mn><mi>t<\/mi><mo>+<\/mo><mn>16<\/mn><msup>\r\n<mi>t<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>&nbsp;\r\n\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>ft, where\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>t<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>t<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is measured in seconds. How long will it take for the object to travel 74 ft?\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2413468\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2413469\" 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-id2413468-solution\">57<\/a><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2413470\">A vacant lot is being converted into a community garden. The garden and the walkway around its perimeter have an area of 378 ft<sup>2<\/sup>. Find the width of the walkway if the garden is 12 ft. wide by 15 ft. long.<\/p>\r\n<span id=\"fs-id2413475\" data-type=\"media\" data-alt=\"A rectangle inside of a larger rectangle. The smaller rectangle has the length labeled: 15 feet and the width labeled: 12 feet. The distance between the two rectangles is labeled as x on all four sides.\">\r\n<img src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"A rectangle inside of a larger rectangle. The smaller rectangle has the length labeled: 15 feet and the width labeled: 12 feet. The distance between the two rectangles is labeled as x on all four sides.\" width=\"378\" height=\"300\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/884eec738c4dd93b9df83bfe221319139f5376fe\" \/>\r\n<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id2385543\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id2385544\" data-type=\"problem\"><span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span>\r\n<div class=\"os-problem-container\">\r\n<p id=\"fs-id2385545\">An epidemiological study of the spread of a certain influenza strain that hit a small school population found that the total number of students,<\/p>\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>P<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>P<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>, who contracted the flu\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>t<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>t<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>days after it broke out is given by the model\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>P<\/mi><mo>=<\/mo><mo>\u2212<\/mo><msup>\r\n<mi>t<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>13<\/mn><mi>t<\/mi><mo>+<\/mo><mn>130<\/mn><mo>,<\/mo>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>P<\/mi><mo>=<\/mo><mo>\u2212<\/mo><msup>\r\n<mi>t<\/mi>\r\n<mn>2<\/mn>\r\n<\/msup>\r\n<mo>+<\/mo><mn>13<\/mn><mi>t<\/mi><mo>+<\/mo><mn>130<\/mn><mo>,<\/mo>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>where\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mn>1<\/mn><mo>\u2264<\/mo><mi>t<\/mi><mo>\u2264<\/mo><mn>6.<\/mn>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mn>1<\/mn><mo>\u2264<\/mo><mi>t<\/mi><mo>\u2264<\/mo><mn>6.<\/mn>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>Find the day that 160 students had the flu. Recall that the restriction on\r\n\r\n<math display=\"inline\"><semantics><mrow>\r\n<mrow>\r\n<mi>t<\/mi>\r\n<\/mrow>\r\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\r\n<mi>t<\/mi>\r\n<\/mrow><\/annotation-xml><\/semantics><\/math>is at most 6.\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_00138001-23fe-4ec9-b2c6-000f3f28ee23\" 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 quadratic equations by factoring.<\/li>\n<li>Solve quadratic equations by the square root property.<\/li>\n<li>Solve quadratic equations by completing the square.<\/li>\n<li>Solve quadratic equations by using the quadratic formula.<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<div id=\"Figure_02_05_001\" class=\"os-figure\">\n<figure class=\"medium\" data-id=\"Figure_02_05_001\"><span id=\"fs-id1278656\" data-type=\"media\" data-alt=\"Two televisions side-by-side. The right television is slightly larger than the left.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Two televisions side-by-side. The right television is slightly larger than the left.\" width=\"655\" height=\"273\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/fa20f121f369fdc2ecdc21a016fa21fe596806bf\" \/><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<p id=\"fs-id1087569\">The computer monitor on the left in <a class=\"autogenerated-content\" href=\"2-5-quadratic-equations#Figure_02_05_001\">Figure 1<\/a> is a 23.6-inch model and the one on the right is a 27-inch model. Proportionally, the monitors appear very similar. If there is a limited amount of space and we desire the largest monitor possible, how do we decide which one to choose? In this section, we will learn how to solve problems such as this using four different methods.<\/p>\n<section id=\"fs-id3182628\" data-depth=\"1\">\n<h2 data-type=\"title\">Solving Quadratic Equations by Factoring<\/h2>\n<p id=\"fs-id2980280\">An equation containing a second-degree polynomial is called a <span id=\"term-00001\" class=\"no-emphasis\" data-type=\"term\">quadratic equation<\/span>. For example, equations such as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\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<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>are quadratic equations. They are used in countless ways in the fields of engineering, architecture, finance, biological science, and, of course, mathematics.<\/p>\n<p id=\"fs-id1385706\">Often the easiest method of solving a quadratic equation is <span id=\"term-00002\" class=\"no-emphasis\" data-type=\"term\">factoring<\/span>. Factoring means finding expressions that can be multiplied together to give the expression on one side of the equation.<\/p>\n<p id=\"fs-id2521760\">If a quadratic equation can be factored, it is written as a product of linear terms. Solving by factoring depends on the zero-product property, which states that if<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>\u22c5<\/mo><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>\u22c5<\/mo><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>then<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>=<\/mo><mn>0<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>or<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>b<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>b<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>where <em data-effect=\"italics\">a <\/em>and <em data-effect=\"italics\">b <\/em>are real numbers or algebraic expressions. In other words, if the product of two numbers or two expressions equals zero, then one of the numbers or one of the expressions must equal zero because zero multiplied by anything equals zero.<\/p>\n<p id=\"fs-id1338084\">Multiplying the factors expands the equation to a string of terms separated by plus or minus signs. So, in that sense, the operation of multiplication undoes the operation of factoring. For example, expand the factored expression<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow><\/annotation-xml><\/semantics><\/math>by multiplying the two factors together.<\/p>\n<div id=\"fs-id2921580\" 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<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/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<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/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<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/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<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1280231\">The product is a quadratic expression. Set equal to zero,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is a quadratic equation. If we were to factor the equation, we would get back the factors we multiplied.<\/p>\n<p id=\"fs-id2996324\">The process of factoring a quadratic equation depends on the leading coefficient, whether it is 1 or another integer. We will look at both situations; but first, we want to confirm that the equation is written in standard form,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>where <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are real numbers, and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>\u2260<\/mo><mn>0.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>\u2260<\/mo><mn>0.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>The equation<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/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<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>is in standard form.<\/p>\n<p id=\"fs-id3176714\">We can use the zero-product property to solve quadratic equations in which we first have to factor out the <span id=\"term-00003\" class=\"no-emphasis\" data-type=\"term\">greatest common factor<\/span> (GCF), and for equations that have special factoring formulas as well, such as the difference of squares, both of which we will see later in this section.<\/p>\n<div id=\"fs-id1318343\" 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=\"\">The Zero-Product Property and Quadratic Equations <\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id2437408\">The <span id=\"term-00004\" data-type=\"term\">zero-product property<\/span> states<\/p>\n<div id=\"fs-id3130490\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtext>If\u00a0<\/mtext><mi>a<\/mi><mo>\u22c5<\/mo><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mtext>then\u00a0<\/mtext><mi>a<\/mi><mo>=<\/mo><mn>0<\/mn><mspace width=\"0.5em\"><\/mspace><mtext>or\u00a0<\/mtext><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtext>If\u00a0<\/mtext><mi>a<\/mi><mo>\u22c5<\/mo><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo><mspace width=\"0.5em\"><\/mspace><mtext>then\u00a0<\/mtext><mi>a<\/mi><mo>=<\/mo><mn>0<\/mn><mspace width=\"0.5em\"><\/mspace><mtext>or\u00a0<\/mtext><mi>b<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2001660\">where <em data-effect=\"italics\">a <\/em>and <em data-effect=\"italics\">b <\/em>are real numbers or algebraic expressions.<\/p>\n<p id=\"fs-id1297051\">A <span id=\"term-00005\" data-type=\"term\">quadratic equation<\/span> is an equation containing a second-degree polynomial; for example<\/p>\n<div id=\"fs-id2714980\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1274659\">where <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are real numbers, and if<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>\u2260<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>\u2260<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>it is in standard form.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1752173\" data-depth=\"2\">\n<h3 data-type=\"title\">Solving Quadratics with a Leading Coefficient of 1<\/h3>\n<p id=\"fs-id2049812\">In the quadratic equation<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>the leading coefficient, or the coefficient of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is 1. We have one method of factoring quadratic equations in this form.<\/p>\n<div id=\"fs-id2048827\" 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-id1279785\"><strong>Given a quadratic equation with the leading coefficient of 1, factor it.<\/strong><\/p>\n<ol id=\"fs-id2436584\" type=\"1\">\n<li>Find two numbers whose product equals <em data-effect=\"italics\">c<\/em> and whose sum equals <em data-effect=\"italics\">b<\/em>.<\/li>\n<li>Use those numbers to write two factors of the form<br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mi>k<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mspace width=\"0.5em\"><\/mspace><mtext>or\u00a0<\/mtext><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mi>k<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mi>k<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mspace width=\"0.5em\"><\/mspace><mtext>or\u00a0<\/mtext><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mi>k<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>where <em data-effect=\"italics\">k <\/em>is one of the numbers found in step 1. Use the numbers exactly as they are. In other words, if the two numbers are 1 and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22122<\/mn><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22122<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>the factors are<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li>Solve using the zero-product property by setting each factor equal to zero and solving for the variable.<\/li>\n<\/ol>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_05_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-id1315053\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2697997\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Factoring and Solving a Quadratic with Leading Coefficient of 1<\/h3>\n<p id=\"fs-id1207622\">Factor and solve the equation:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/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<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id2508977\" 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-id1815377\">To factor<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>we look for two numbers whose product equals<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22126<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22126<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and whose sum equals 1. Begin by looking at the possible factors of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22126.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22126.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id2771168\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd>\n<mrow>\n<mn>1<\/mn><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22126<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>\u22126<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd>\n<mrow>\n<mn>2<\/mn><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22123<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd>\n<mrow>\n<mn>3<\/mn><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">)<\/mo><\/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<mn>1<\/mn><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22126<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>\u22126<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd>\n<mrow>\n<mn>2<\/mn><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22123<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd>\n<mrow>\n<mn>3<\/mn><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22122<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1722929\">The last pair,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><mo>\u22c5<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22122<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><mo>\u22c5<\/mo><mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22122<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow><\/annotation-xml><\/semantics><\/math>sums to 1, so these are the numbers. Note that only one pair of numbers will work. Then, write the factors.<\/p>\n<div id=\"fs-id1466455\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1779896\">To solve this equation, we use the zero-product property. Set each factor equal to zero and solve.<\/p>\n<div id=\"fs-id2437878\" 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<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>0<\/mn><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>0<\/mn><\/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>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>0<\/mn><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mrow><mn>\u22123<\/mn><\/mrow><\/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<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\"><mn>0<\/mn><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>0<\/mn><\/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>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>0<\/mn><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mrow><mn>\u22123<\/mn><\/mrow><\/mtd>\n<\/mtr>\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2753851\">The two solutions are<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22123.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22123.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>We can see how the solutions relate to the graph in <a class=\"autogenerated-content\" href=\"2-5-quadratic-equations#Figure_02_05_002\">Figure 2<\/a>. The solutions are the <em data-effect=\"italics\">x-<\/em>intercepts of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow><mi>y<\/mi>\n<mo>=<\/mo><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow><mi>y<\/mi>\n<mo>=<\/mo><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>6<\/mn><mo>=<\/mo><mn>0.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"Figure_02_05_002\" class=\"os-figure\">\n<figure class=\"small\" data-id=\"Figure_02_05_002\"><span id=\"fs-id2381086\" data-type=\"media\" data-alt=\"Coordinate plane with the x-axis ranging from negative 5 to 5 and the y-axis ranging from negative 7 to 7. The function x squared plus x minus six equals zero is graphed, with the x-intercepts (-3,0) and (2,0), plotted as well.\"><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 7 to 7. The function x squared plus x minus six equals zero is graphed, with the x-intercepts (-3,0) and (2,0), plotted as well.\" width=\"487\" height=\"588\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/d245f4bb5c2445c5f7b842f335d44530209b28d6\" \/><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<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1510352\" 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_05_01\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1769680\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id2294713\">Factor and solve the quadratic equation:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/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<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/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>\n<\/section>\n<\/div>\n<div id=\"Example_02_05_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-id3052323\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1996793\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solve the Quadratic Equation by Factoring<\/h3>\n<p id=\"fs-id1045593\">Solve the quadratic equation by factoring:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>8<\/mn><mi>x<\/mi><mo>+<\/mo><mn>15<\/mn><mo>=<\/mo><mn>0.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>8<\/mn><mi>x<\/mi><mo>+<\/mo><mn>15<\/mn><mo>=<\/mo><mn>0.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1834290\" 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-id1690316\">Find two numbers whose product equals<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>15<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>15<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and whose sum equals<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>8.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>8.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>List the factors of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>15.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>15.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id2892822\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"left\"><mrow><mn>1<\/mn><mo>\u22c5<\/mo><mn>15<\/mn><\/mrow><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"left\"><mrow><mn>3<\/mn><mo>\u22c5<\/mo><mn>5<\/mn><\/mrow><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"left\"><mrow><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u221215<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"left\"><mrow><mo stretchy=\"false\">(<\/mo><mn>\u22123<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22125<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"left\"><mrow><mn>1<\/mn><mo>\u22c5<\/mo><mn>15<\/mn><\/mrow><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"left\"><mrow><mn>3<\/mn><mo>\u22c5<\/mo><mn>5<\/mn><\/mrow><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"left\"><mrow><mo stretchy=\"false\">(<\/mo><mn>\u22121<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u221215<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"left\"><mrow><mo stretchy=\"false\">(<\/mo><mn>\u22123<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>\u22125<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2629248\">The numbers that add to 8 are 3 and 5. Then, write the factors, set each factor equal to zero, and solve.<\/p>\n<div id=\"fs-id2364916\" 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<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/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<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/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<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<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/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<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>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/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<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/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<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<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>5<\/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<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>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1288611\">The solutions are<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22123<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22123<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/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>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2431514\" 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_05_02\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1515965\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1545876\">Solve the quadratic equation by factoring:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>21<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>x<\/mi><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<div id=\"Example_02_05_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-id1442204\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1766773\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Using the Zero-Product Property to Solve a Quadratic Equation Written as the Difference of Squares<\/h3>\n<p id=\"fs-id1007551\">Solve the difference of squares equation using the zero-product property:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<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>\n<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<details id=\"fs-id1467645\" 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-id1342314\">Recognizing that the equation represents the difference of squares, we can write the two factors by taking the square root of each term, using a minus sign as the operator in one factor and a plus sign as the operator in the other. Solve using the zero-factor property.<\/p>\n<div id=\"fs-id1723089\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\"><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>9<\/mn><\/mrow><\/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<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/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><mrow><mo>(<\/mo><mrow><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow><mo>)<\/mo><\/mrow><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>0<\/mn><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>3<\/mn><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mrow><mo>(<\/mo><mrow><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow><mo>)<\/mo><\/mrow><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>0<\/mn><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mrow><mn>\u22123<\/mn><\/mrow><\/mtd>\n<\/mtr>\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\"><mrow><msup><mi>x<\/mi><mn>2<\/mn><\/msup><mo>\u2212<\/mo><mn>9<\/mn><\/mrow><\/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<mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/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><mrow><mo>(<\/mo><mrow><mi>x<\/mi><mo>\u2212<\/mo><mn>3<\/mn><\/mrow><mo>)<\/mo><\/mrow><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>0<\/mn><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>3<\/mn><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mrow><mo>(<\/mo><mrow><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow><mo>)<\/mo><\/mrow><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>0<\/mn><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mrow><mn>\u22123<\/mn><\/mrow><\/mtd>\n<\/mtr>\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1759726\">The solutions are<\/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>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22123.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22123.<\/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-id2029177\" 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_05_03\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1322224\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id2751610\">Solve by factoring:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>25<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>25<\/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-id2876101\" data-depth=\"2\">\n<h3 data-type=\"title\">Solving a Quadratic Equation by Factoring when the Leading Coefficient is not 1<\/h3>\n<p id=\"fs-id2802890\">When the leading coefficient is not 1, we factor a quadratic equation using the method called grouping, which requires four terms. With the equation in standard form, let\u2019s review the grouping procedures:<\/p>\n<ol id=\"fs-id1445659\" type=\"1\">\n<li>With the quadratic in standard form,<br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>multiply<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>\u22c5<\/mo><mi>c<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>\u22c5<\/mo><mi>c<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li>Find two numbers whose product equals<br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mi>c<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mi>c<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and whose sum equals<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>b<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>b<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li>Rewrite the equation replacing the<br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>b<\/mi><mi>x<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>b<\/mi><mi>x<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>term with two terms using the numbers found in step 2 as coefficients of <em data-effect=\"italics\">x.<\/em><\/li>\n<li>Factor the first two terms and then factor the last two terms. The expressions in parentheses must be exactly the same to use grouping.<\/li>\n<li>Factor out the expression in parentheses.<\/li>\n<li>Set the expressions equal to zero and solve for the variable.<\/li>\n<\/ol>\n<div id=\"Example_02_05_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-id2917470\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2505105\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving a Quadratic Equation Using Grouping<\/h3>\n<p id=\"fs-id2638849\">Use grouping to factor and solve the quadratic equation:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>4<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>15<\/mn><mi>x<\/mi><mo>+<\/mo><mn>9<\/mn><mo>=<\/mo><mn>0.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>4<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>15<\/mn><mi>x<\/mi><mo>+<\/mo><mn>9<\/mn><mo>=<\/mo><mn>0.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1223661\" 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-id1558604\">First, multiply<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mi>c<\/mi><mo>:<\/mo><mn>4<\/mn><mrow><mo>(<\/mo>\n<mn>9<\/mn>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>36.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mi>c<\/mi><mo>:<\/mo><mn>4<\/mn><mrow><mo>(<\/mo>\n<mn>9<\/mn>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>36.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>Then list the factors of<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>36.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>36.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<div id=\"fs-id1449438\" 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<mn>1<\/mn><mo>\u22c5<\/mo><mn>36<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>2<\/mn><mo>\u22c5<\/mo><mn>18<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>3<\/mn><mo>\u22c5<\/mo><mn>12<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>4<\/mn><mo>\u22c5<\/mo><mn>9<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>6<\/mn><mo>\u22c5<\/mo><mn>6<\/mn>\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<mn>1<\/mn><mo>\u22c5<\/mo><mn>36<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>2<\/mn><mo>\u22c5<\/mo><mn>18<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>3<\/mn><mo>\u22c5<\/mo><mn>12<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>4<\/mn><mo>\u22c5<\/mo><mn>9<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mn>6<\/mn><mo>\u22c5<\/mo><mn>6<\/mn>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id3107108\">The only pair of factors that sums to<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>15<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>15<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><mo>+<\/mo><mn>12.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><mo>+<\/mo><mn>12.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>Rewrite the equation replacing the <em data-effect=\"italics\">b <\/em>term,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>15<\/mn><mi>x<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>15<\/mn><mi>x<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>with two terms using 3 and 12 as coefficients of <em data-effect=\"italics\">x<\/em>. Factor the first two terms, and then factor the last two terms.<\/p>\n<div id=\"fs-id1417674\" 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>4<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>12<\/mn><mi>x<\/mi><mo>+<\/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<mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/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><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/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<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>4<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>12<\/mn><mi>x<\/mi><mo>+<\/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<mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/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><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/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<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1579301\">Solve using the zero-product property.<\/p>\n<div id=\"fs-id2502349\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>0<\/mn><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>0<\/mn><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>0<\/mn><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mrow><mo>\u2212<\/mo><mn>3<\/mn><\/mrow><\/mtd>\n<\/mtr>\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>0<\/mn><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>0<\/mn><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mrow><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>0<\/mn><\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\"><mi>x<\/mi><\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mrow><mo>\u2212<\/mo><mn>3<\/mn><\/mrow><\/mtd>\n<\/mtr>\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2432037\">The solutions are<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>, <\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<mo>, <\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22123.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22123.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>See <a class=\"autogenerated-content\" href=\"2-5-quadratic-equations#Figure_02_05_003\">Figure 3<\/a>.<\/p>\n<div id=\"Figure_02_05_003\" class=\"os-figure\">\n<figure class=\"small\" data-id=\"Figure_02_05_003\"><span id=\"fs-id1269396\" data-type=\"media\" data-alt=\"Coordinate plane with the x-axis ranging from negative 6 to 2 with every other tick mark labeled and the y-axis ranging from negative 6 to 2 with each tick mark numbered. The equation: four x squared plus fifteen x plus nine is graphed with its x-intercepts: (-3\/4,0) and (-3,0) plotted as well.\"><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 6 to 2 with every other tick mark labeled and the y-axis ranging from negative 6 to 2 with each tick mark numbered. The equation: four x squared plus fifteen x plus nine is graphed with its x-intercepts: (-3\/4,0) and (-3,0) plotted as well.\" width=\"487\" height=\"433\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/c8c54dba0d1717875901655bddf937a8b9e3db6a\" \/><br \/>\n<\/span><\/figure>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span><br \/>\n<span class=\"os-number\">3<\/span><\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1554615\" 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_05_04\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2629095\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id2697369\">Solve using factoring by grouping:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>12<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>11<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>12<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>11<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0.<\/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_05_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-id3115239\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1561758\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving a Polynomial of Higher Degree by Factoring<\/h3>\n<p id=\"fs-id3182370\">Solve the equation by factoring:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22123<\/mn><msup>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>5<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>=<\/mo><mn>0.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22123<\/mn><msup>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>5<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><mo>=<\/mo><mn>0.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1400268\" 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-id2440018\">This equation does not look like a quadratic, as the highest power is 3, not 2. Recall that the first thing we want to do when solving any equation is to factor out the GCF, if one exists. And it does here. We can factor out<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo>\u2212<\/mo><mi>x<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>\u2212<\/mo><mi>x<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>from all of the terms and then proceed with grouping.<\/p>\n<div id=\"fs-id2367474\" 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>\u22123<\/mn><msup>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>5<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/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<mo>\u2212<\/mo><mi>x<\/mi><mo>(<\/mo><mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>)<\/mo><\/mrow>\n<\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>0<\/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>\u22123<\/mn><msup>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>5<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>2<\/mn><mi>x<\/mi><\/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<mo>\u2212<\/mo><mi>x<\/mi><mo>(<\/mo><mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>)<\/mo><\/mrow>\n<\/mtd>\n<mtd><mo>=<\/mo><\/mtd>\n<mtd columnalign=\"left\"><mn>0<\/mn><\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1752856\">Use grouping on the expression in parentheses.<\/p>\n<div id=\"fs-id2869363\" 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>\u2212<\/mo><mi>x<\/mi><mo>(<\/mo><mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/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<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mo>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">[<\/mo><mn>3<\/mn><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><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>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/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>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mo>\u2212<\/mo><mi>x<\/mi><mo>(<\/mo><mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/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<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mo>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">[<\/mo><mn>3<\/mn><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><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>\u2212<\/mo><mi>x<\/mi><mo stretchy=\"false\">(<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/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>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1552054\">Now, we use the zero-product property. Notice that we have three factors.<\/p>\n<div id=\"fs-id1200475\" 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>\u2212<\/mo><mi>x<\/mi><\/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<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/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>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>x<\/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<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<mo>\u2212<\/mo><mi>x<\/mi><\/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<mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/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>x<\/mi>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>x<\/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<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-id2655312\">The solutions are<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>0<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>0<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>2<\/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>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<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-id2500651\" 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_05_05\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1254757\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1254758\">Solve by factoring:<\/p>\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>11<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>10<\/mn><mi>x<\/mi><mo>=<\/mo><mn>0.<\/mn>\n<\/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>11<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>10<\/mn><mi>x<\/mi><mo>=<\/mo><mn>0.<\/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>\n<section id=\"fs-id1402144\" data-depth=\"1\">\n<h2 data-type=\"title\">Using the Square Root Property<\/h2>\n<p id=\"fs-id1923792\">When there is no linear term in the equation, another method of solving a quadratic equation is by using the <span id=\"term-00006\" data-type=\"term\">square root property<\/span>, in which we isolate the<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>term and take the square root of the number on the other side of the equals sign. Keep in mind that sometimes we may have to manipulate the equation to isolate the<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>term so that the square root property can be used.<\/p>\n<div id=\"fs-id1831215\" 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=\"\">The Square Root Property<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1569584\">With the<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>term isolated, the square root property states that:<\/p>\n<div id=\"fs-id1499182\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mtext>if<\/mtext><mspace width=\"0.5em\"><\/mspace><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mi>k<\/mi><mo>,<\/mo><mtext>then<\/mtext><mspace width=\"0.5em\"><\/mspace><mi>x<\/mi><mo>=<\/mo><mo>\u00b1<\/mo><msqrt>\n<mi>k<\/mi>\n<\/msqrt>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtext>if<\/mtext><mspace width=\"0.5em\"><\/mspace><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mi>k<\/mi><mo>,<\/mo><mtext>then<\/mtext><mspace width=\"0.5em\"><\/mspace><mi>x<\/mi><mo>=<\/mo><mo>\u00b1<\/mo><msqrt>\n<mi>k<\/mi>\n<\/msqrt>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2875920\">where <em data-effect=\"italics\">k <\/em>is a nonzero real number.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1521248\" 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-id1155370\"><strong>Given a quadratic equation with an <\/strong><\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>term but no<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>term, use the square root property to solve it.<\/p>\n<ol id=\"fs-id3081167\" type=\"1\">\n<li>Isolate the<br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>term on one side of the equal sign.<\/li>\n<li>Take the square root of both sides of the equation, putting a<br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo>\u00b1<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>\u00b1<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>sign before the expression on the side opposite the squared term.<\/li>\n<li>Simplify the numbers on the side with the<br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo>\u00b1<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>\u00b1<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>sign.<\/li>\n<\/ol>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_05_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-id1905142\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1579057\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving a Simple Quadratic Equation Using the Square Root Property<\/h3>\n<p id=\"fs-id1467277\">Solve the quadratic using the square root property:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>8.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>8.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1514577\" 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-id1515767\">Take the square root of both sides, and then simplify the radical. Remember to use a<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo>\u00b1<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>\u00b1<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>sign before the radical symbol.<\/p>\n<div id=\"fs-id2437570\" 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<\/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<\/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<mo>\u00b1<\/mo><msqrt>\n<mn>8<\/mn>\n<\/msqrt>\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\">\n<mrow>\n<mo>\u00b1<\/mo><mn>2<\/mn><msqrt>\n<mn>2<\/mn>\n<\/msqrt>\n<\/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<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/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<\/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<mo>\u00b1<\/mo><msqrt>\n<mn>8<\/mn>\n<\/msqrt>\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\">\n<mrow>\n<mo>\u00b1<\/mo><mn>2<\/mn><msqrt>\n<mn>2<\/mn>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2437670\">The solutions are<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><msqrt>\n<mn>2<\/mn>\n<\/msqrt>\n<mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><msqrt>\n<mn>2<\/mn>\n<\/msqrt>\n<mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22122<\/mn><msqrt>\n<mn>2<\/mn>\n<\/msqrt>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22122<\/mn><msqrt>\n<mn>2<\/mn>\n<\/msqrt>\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=\"Example_02_05_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-id862941\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1340166\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving a Quadratic Equation Using the Square Root Property<\/h3>\n<p id=\"fs-id2497523\">Solve the quadratic equation:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>4<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mtext>7.<\/mtext><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>4<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mtext>7.<\/mtext><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1482615\" 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-id1568725\">First, isolate the<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>term. Then take the square root of both sides.<\/p>\n<div id=\"fs-id2317298\" 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>4<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>7<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>4<\/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>6<\/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<mrow>\n<mfrac>\n<mn>6<\/mn>\n<mn>4<\/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><mfrac>\n<mrow>\n<msqrt>\n<mn>6<\/mn>\n<\/msqrt>\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\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>4<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>1<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>7<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mn>4<\/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>6<\/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<mrow>\n<mfrac>\n<mn>6<\/mn>\n<mn>4<\/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><mfrac>\n<mrow>\n<msqrt>\n<mn>6<\/mn>\n<\/msqrt>\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1522278\">The solutions are<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<msqrt>\n<mn>6<\/mn>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>, <\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<msqrt>\n<mn>6<\/mn>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<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<mrow>\n<msqrt>\n<mn>6<\/mn>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>\u2212<\/mo><mfrac>\n<mrow>\n<msqrt>\n<mn>6<\/mn>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mn>2<\/mn>\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-id1257773\" 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_05_06\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1913154\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id1913155\">Solve the quadratic equation using the square root property:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>15.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>15.<\/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-id1956704\" data-depth=\"1\">\n<h2 data-type=\"title\">Completing the Square<\/h2>\n<p id=\"fs-id1793627\">Not all quadratic equations can be factored or can be solved in their original form using the square root property. In these cases, we may use a method for solving a <span id=\"term-00007\" class=\"no-emphasis\" data-type=\"term\">quadratic equation<\/span> known as <span id=\"term-00008\" data-type=\"term\">completing the square<\/span>. Using this method, we add or subtract terms to both sides of the equation until we have a perfect square trinomial on one side of the equal sign. We then apply the square root property. To complete the square, the leading coefficient, <em data-effect=\"italics\">a<\/em>, must equal 1. If it does not, then divide the entire equation by <em data-effect=\"italics\">a<\/em>. Then, we can use the following procedures to solve a quadratic equation by completing the square.<\/p>\n<p id=\"fs-id1538204\">We will use the example<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>to illustrate each step.<\/p>\n<ol id=\"fs-id1234074\" type=\"1\">\n<li>\n<p id=\"fs-id1583461\">Given a quadratic equation that cannot be factored, and with<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>first add or subtract the constant term to the right side of the equal sign.<\/p>\n<div id=\"fs-id1512735\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>=<\/mo><mn>\u22121<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>=<\/mo><mn>\u22121<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/li>\n<li>\n<p id=\"fs-id1175291\">Multiply the <em data-effect=\"italics\">b <\/em>term by<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>and square it.<\/p>\n<div id=\"fs-id1410870\" 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<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>2<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msup>\n<mn>2<\/mn>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>4<\/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<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>2<\/mn>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msup>\n<mn>2<\/mn>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mn>4<\/mn>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/li>\n<li>\n<p id=\"fs-id1514197\">Add<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>b<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mi>b<\/mi>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>to both sides of the equal sign and simplify the right side. We have<\/p>\n<div id=\"fs-id1526634\" 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>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mn>1<\/mn><mo>+<\/mo><mn>4<\/mn><\/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<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><\/mrow>\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<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mn>1<\/mn><mo>+<\/mo><mn>4<\/mn><\/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<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><\/mrow>\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<\/li>\n<li>\n<p id=\"fs-id1538635\">The left side of the equation can now be factored as a perfect square.<\/p>\n<div id=\"fs-id1719027\" 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>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><\/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<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\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<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><\/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<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\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<\/li>\n<li>\n<p id=\"fs-id2707455\">Use the square root property and solve.<\/p>\n<div id=\"fs-id2385243\" 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<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u00b1<\/mo><msqrt>\n<mn>3<\/mn>\n<\/msqrt>\n<\/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<mo>\u00b1<\/mo><msqrt>\n<mn>3<\/mn>\n<\/msqrt>\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<mn>\u22122<\/mn><mo>\u00b1<\/mo><msqrt>\n<mn>3<\/mn>\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<msqrt>\n<mrow>\n<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u00b1<\/mo><msqrt>\n<mn>3<\/mn>\n<\/msqrt>\n<\/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<mo>\u00b1<\/mo><msqrt>\n<mn>3<\/mn>\n<\/msqrt>\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<mn>\u22122<\/mn><mo>\u00b1<\/mo><msqrt>\n<mn>3<\/mn>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/li>\n<li>\n<p id=\"fs-id3263921\">The solutions are<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22122<\/mn><mo>+<\/mo><msqrt>\n<mn>3<\/mn>\n<\/msqrt>\n<mo>,<\/mo><mo> <\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22122<\/mn><mo>+<\/mo><msqrt>\n<mn>3<\/mn>\n<\/msqrt>\n<mo>,<\/mo><mo> <\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>\u22122<\/mn><mo>\u2212<\/mo><msqrt>\n<mn>3<\/mn>\n<\/msqrt>\n<mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>\u22122<\/mn><mo>\u2212<\/mo><msqrt>\n<mn>3<\/mn>\n<\/msqrt>\n<mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/li>\n<\/ol>\n<div id=\"Example_02_05_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-id1884309\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1199291\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving a Quadratic by Completing the Square<\/h3>\n<p id=\"fs-id2979951\">Solve the quadratic equation by completing the square:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id2933237\" 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-id2523221\">First, move the constant term to the right side of the equal sign.<\/p>\n<div id=\"fs-id1560122\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><mo>=<\/mo><mn>5<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1701880\">Then, take<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>of the <em data-effect=\"italics\">b <\/em>term and square it.<\/p>\n<div id=\"fs-id1700237\" 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<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo stretchy=\"false\">(<\/mo><mn>\u22123<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\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<mfrac>\n<mn>9<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\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<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo stretchy=\"false\">(<\/mo><mn>\u22123<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\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<mfrac>\n<mn>9<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id3116426\">Add the result to both sides of the equal sign.<\/p>\n<div id=\"fs-id1919626\" 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>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\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<mn>5<\/mn><mo>+<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\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<mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mfrac>\n<mn>9<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>5<\/mn><mo>+<\/mo><mfrac>\n<mn>9<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\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<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\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<mn>5<\/mn><mo>+<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\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<mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><mo>+<\/mo><mfrac>\n<mn>9<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mn>5<\/mn><mo>+<\/mo><mfrac>\n<mn>9<\/mn>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2495836\">Factor the left side as a perfect square and simplify the right side.<\/p>\n<div id=\"fs-id3176685\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mfrac>\n<mrow>\n<mn>29<\/mn>\n<\/mrow>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/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><mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mfrac>\n<mrow>\n<mn>29<\/mn>\n<\/mrow>\n<mn>4<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1482511\">Use the square root property and solve.<\/p>\n<div id=\"fs-id1545239\" 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<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u00b1<\/mo><msqrt>\n<mrow>\n<mfrac>\n<mrow>\n<mn>29<\/mn><\/mrow>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u00b1<\/mo><mfrac>\n<mrow>\n<msqrt>\n<mrow>\n<mn>29<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mn>2<\/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<mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>\u00b1<\/mo><mfrac>\n<mrow>\n<msqrt>\n<mrow>\n<mn>29<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\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<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u00b1<\/mo><msqrt>\n<mrow>\n<mfrac>\n<mrow>\n<mn>29<\/mn><\/mrow>\n<mn>4<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u00b1<\/mo><mfrac>\n<mrow>\n<msqrt>\n<mrow>\n<mn>29<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mn>2<\/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<mfrac>\n<mn>3<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mo>\u00b1<\/mo><mfrac>\n<mrow>\n<msqrt>\n<mrow>\n<mn>29<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id2628693\">The solutions are<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mfrac>\n<mrow>\n<mn>3<\/mn><mo>+<\/mo>\n<msqrt>\n<mn>29<\/mn>\n<\/msqrt>\n<\/mrow>\n<mrow>\n<mn>2<\/mn>\n<\/mrow>\n<\/mfrac>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mfrac>\n<mrow>\n<mn>3<\/mn><mo>+<\/mo>\n<msqrt>\n<mn>29<\/mn>\n<\/msqrt>\n<\/mrow>\n<mrow>\n<mn>2<\/mn>\n<\/mrow>\n<\/mfrac><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mfrac>\n<mrow>\n<mn>3<\/mn><mo>&#8211;<\/mo>\n<msqrt>\n<mn>29<\/mn>\n<\/msqrt>\n<\/mrow>\n<mrow>\n<mn>2<\/mn>\n<\/mrow>\n<\/mfrac>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mfrac>\n<mrow>\n<mn>3<\/mn><mo>&#8211;<\/mo>\n<msqrt>\n<mn>29<\/mn>\n<\/msqrt>\n<\/mrow>\n<mrow>\n<mn>2<\/mn>\n<\/mrow>\n<\/mfrac><\/annotation-xml><\/semantics><\/math>.<\/p>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2410455\" 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_05_07\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3086043\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id3086044\">Solve by completing the square:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>6<\/mn><mi>x<\/mi><mo>=<\/mo><mn>13.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>6<\/mn><mi>x<\/mi><mo>=<\/mo><mn>13.<\/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-id2979976\" data-depth=\"1\">\n<h2 data-type=\"title\">Using the Quadratic Formula<\/h2>\n<p id=\"fs-id2367499\">The fourth method of solving a <span id=\"term-00009\" class=\"no-emphasis\" data-type=\"term\">quadratic equation<\/span> is by using the <span id=\"term-00010\" class=\"no-emphasis\" data-type=\"term\">quadratic formula<\/span>, a formula that will solve all quadratic equations. Although the quadratic formula works on any quadratic equation in standard form, it is easy to make errors in substituting the values into the formula. Pay close attention when substituting, and use parentheses when inserting a negative number.<\/p>\n<p id=\"fs-id1335013\">We can derive the quadratic formula by <span id=\"term-00011\" class=\"no-emphasis\" data-type=\"term\">completing the square<\/span>. We will assume that the leading coefficient is positive; if it is negative, we can multiply the equation by<\/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>and obtain a positive <em data-effect=\"italics\">a<\/em>. Given<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>\u2260<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>\u2260<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>we will complete the square as follows:<\/p>\n<ol id=\"fs-id1894962\" type=\"1\">\n<li>\n<p id=\"fs-id2429906\">First, move the constant term to the right side of the equal sign:<\/p>\n<div id=\"fs-id1547656\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>c<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mi>c<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/li>\n<li>\n<p id=\"fs-id1569987\">As we want the leading coefficient to equal 1, divide through by <em data-effect=\"italics\">a<\/em>:<\/p>\n<div id=\"fs-id1528274\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mfrac>\n<mi>b<\/mi>\n<mi>a<\/mi>\n<\/mfrac>\n<mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mi>c<\/mi>\n<mi>a<\/mi>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mfrac>\n<mi>b<\/mi>\n<mi>a<\/mi>\n<\/mfrac>\n<mi>x<\/mi><mo>=<\/mo><mo>\u2212<\/mo><mfrac>\n<mi>c<\/mi>\n<mi>a<\/mi>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/li>\n<li>\n<p id=\"fs-id2906392\">Then, find<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>of the middle term, and add<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mfrac>\n<mi>b<\/mi>\n<mi>a<\/mi>\n<\/mfrac>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mfrac>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<mrow>\n<mn>4<\/mn><msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mfrac>\n<mn>1<\/mn>\n<mn>2<\/mn>\n<\/mfrac>\n<mfrac>\n<mi>b<\/mi>\n<mi>a<\/mi>\n<\/mfrac>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mfrac>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<mrow>\n<mn>4<\/mn><msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>to both sides of the equal sign:<\/p>\n<div id=\"fs-id2443764\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mfrac>\n<mi>b<\/mi>\n<mi>a<\/mi>\n<\/mfrac>\n<mi>x<\/mi><mo>+<\/mo><mfrac>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<mrow>\n<mn>4<\/mn><msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<mrow>\n<mn>4<\/mn><msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mi>c<\/mi>\n<mi>a<\/mi>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mfrac>\n<mi>b<\/mi>\n<mi>a<\/mi>\n<\/mfrac>\n<mi>x<\/mi><mo>+<\/mo><mfrac>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<mrow>\n<mn>4<\/mn><msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mfrac>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<mrow>\n<mn>4<\/mn><msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mi>c<\/mi>\n<mi>a<\/mi>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/li>\n<li>\n<p id=\"fs-id1719390\">Next, write the left side as a perfect square. Find the common denominator of the right side and write it as a single fraction:<\/p>\n<div id=\"fs-id3148450\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mfrac>\n<mi>b<\/mi>\n<mrow>\n<mn>2<\/mn><mi>a<\/mi>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mfrac>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi>\n<\/mrow>\n<mrow>\n<mn>4<\/mn><msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/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><mfrac>\n<mi>b<\/mi>\n<mrow>\n<mn>2<\/mn><mi>a<\/mi>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mfrac>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi>\n<\/mrow>\n<mrow>\n<mn>4<\/mn><msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/li>\n<li>\n<p id=\"fs-id2307686\">Now, use the square root property, which gives<\/p>\n<div id=\"fs-id1937587\" 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>x<\/mi><mo>+<\/mo><mfrac>\n<mi>b<\/mi>\n<mrow>\n<mn>2<\/mn><mi>a<\/mi><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u00b1<\/mo><msqrt>\n<mrow>\n<mfrac>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow>\n<mrow>\n<mn>4<\/mn><msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mfrac>\n<mi>b<\/mi>\n<mrow>\n<mn>2<\/mn><mi>a<\/mi><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u00b1<\/mo><msqrt>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mi>a<\/mi><\/mrow>\n<\/mfrac>\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<mi>x<\/mi><mo>+<\/mo><mfrac>\n<mi>b<\/mi>\n<mrow>\n<mn>2<\/mn><mi>a<\/mi><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mo>\u00b1<\/mo><msqrt>\n<mrow>\n<mfrac>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow>\n<mrow>\n<mn>4<\/mn><msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/msqrt>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd columnalign=\"right\">\n<mrow>\n<mi>x<\/mi><mo>+<\/mo><mfrac>\n<mi>b<\/mi>\n<mrow>\n<mn>2<\/mn><mi>a<\/mi><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<mtd>\n<mo>=<\/mo>\n<\/mtd>\n<mtd columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u00b1<\/mo><msqrt>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mi>a<\/mi><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/li>\n<li>\n<p id=\"fs-id1977996\">Finally, add<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mo>\u2212<\/mo><mfrac>\n<mi>b<\/mi>\n<mrow>\n<mn>2<\/mn><mi>a<\/mi>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mo>\u2212<\/mo><mfrac>\n<mi>b<\/mi>\n<mrow>\n<mn>2<\/mn><mi>a<\/mi>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>to both sides of the equation and combine the terms on the right side. Thus,<\/p>\n<div id=\"fs-id2440094\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>=<\/mo><mfrac>\n<mrow>\n<mo>\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mi>a<\/mi>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>=<\/mo><mfrac>\n<mrow>\n<mo>\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mi>a<\/mi>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<\/li>\n<\/ol>\n<div id=\"fs-id2500151\" 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=\"\">The Quadratic Formula<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1278348\">Written in standard form,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>any quadratic equation can be solved using the <span id=\"term-00012\" data-type=\"term\">quadratic formula<\/span>:<\/p>\n<div id=\"fs-id1959865\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi><mo>=<\/mo><mfrac>\n<mrow>\n<mo>\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mi>a<\/mi>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi><mo>=<\/mo><mfrac>\n<mrow>\n<mo>\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi>\n<\/mrow>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mi>a<\/mi>\n<\/mrow>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1930451\">where <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em>, and <em data-effect=\"italics\">c<\/em> are real numbers and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>\u2260<\/mo><mn>0.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>\u2260<\/mo><mn>0.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1480868\" 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-id2385664\"><strong>Given a quadratic equation, solve it using the quadratic formula<\/strong><\/p>\n<ol id=\"fs-id1973820\" type=\"1\">\n<li>Make sure the equation is in standard form:<br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li>Make note of the values of the coefficients and constant term,<br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>,<\/mo><mi>b<\/mi><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>,<\/mo><mi>b<\/mi><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>c<\/mi><mo>.<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>c<\/mi><mo>.<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li>Carefully substitute the values noted in step 2 into the equation. To avoid needless errors, use parentheses around each number input into the formula.<\/li>\n<li>Calculate and solve.<\/li>\n<\/ol>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_05_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-id2979701\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2979703\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solve the Quadratic Equation Using the Quadratic Formula<\/h3>\n<p id=\"fs-id2442582\">Solve the quadratic equation:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id3034009\" 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-id1800418\">Identify the coefficients:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo><mi>b<\/mi><mo>=<\/mo><mn>5<\/mn><mo>,<\/mo><mi>c<\/mi><mo>=<\/mo><mn>1.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo><mi>b<\/mi><mo>=<\/mo><mn>5<\/mn><mo>,<\/mo><mi>c<\/mi><mo>=<\/mo><mn>1.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>Then use the quadratic formula.<\/p>\n<div id=\"fs-id1813797\" 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<mi>x<\/mi>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u00b1<\/mo><msqrt>\n<mrow>\n<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mfrac>\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\">\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>5<\/mn><mo>\u00b1<\/mo><msqrt>\n<mrow>\n<mn>25<\/mn><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>5<\/mn><mo>\u00b1<\/mo><msqrt>\n<mrow>\n<mn>21<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<\/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<mi>x<\/mi>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u00b1<\/mo><msqrt>\n<mrow>\n<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>5<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/mfrac>\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\">\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>5<\/mn><mo>\u00b1<\/mo><msqrt>\n<mrow>\n<mn>25<\/mn><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>5<\/mn><mo>\u00b1<\/mo><msqrt>\n<mrow>\n<mn>21<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/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=\"Example_02_05_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-id1561272\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1457087\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Solving a Quadratic Equation with the Quadratic Formula<\/h3>\n<p id=\"fs-id1457092\">Use the quadratic formula to solve<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<details id=\"fs-id1717549\" 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-id1717551\">First, we identify the coefficients:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo><mi>b<\/mi><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo><mi>b<\/mi><mo>=<\/mo><mn>1<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>c<\/mi><mo>=<\/mo><mn>2.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>c<\/mi><mo>=<\/mo><mn>2.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p id=\"fs-id1531480\">Substitute these values into the quadratic formula.<\/p>\n<div id=\"fs-id2486075\" 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<mi>x<\/mi>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mi>a<\/mi><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u00b1<\/mo><msqrt>\n<mrow>\n<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><\/mrow>\n<\/mfrac>\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\">\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>1<\/mn><mo>\u00b1<\/mo><msqrt>\n<mrow>\n<mn>1<\/mn><mo>\u2212<\/mo><mn>8<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\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\">\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>1<\/mn><mo>\u00b1<\/mo><msqrt>\n<mrow>\n<mo>\u2212<\/mo><mn>7<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\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\">\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>1<\/mn><mo>\u00b1<\/mo><mi>i<\/mi><msqrt>\n<mn>7<\/mn>\n<\/msqrt>\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<\/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<mi>x<\/mi>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mi>b<\/mi><mo>\u00b1<\/mo><msqrt>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mi>a<\/mi><\/mrow>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr>\n<mtd rowalign=\"center\"><\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u00b1<\/mo><msqrt>\n<mrow>\n<msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>1<\/mn><mo stretchy=\"false\">)<\/mo><mo>\u22c5<\/mo><mo stretchy=\"false\">(<\/mo><mn>2<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mrow>\n<mn>2<\/mn><mo>\u22c5<\/mo><mn>1<\/mn><\/mrow>\n<\/mfrac>\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\">\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>1<\/mn><mo>\u00b1<\/mo><msqrt>\n<mrow>\n<mn>1<\/mn><mo>\u2212<\/mo><mn>8<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\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\">\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>1<\/mn><mo>\u00b1<\/mo><msqrt>\n<mrow>\n<mo>\u2212<\/mo><mn>7<\/mn><\/mrow>\n<\/msqrt>\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\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\">\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>1<\/mn><mo>\u00b1<\/mo><mi>i<\/mi><msqrt>\n<mn>7<\/mn>\n<\/msqrt>\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>\n<\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1798703\">The solutions to the equation are<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>1<\/mn><mo>+<\/mo><mi>i<\/mi><msqrt>\n<mn>7<\/mn>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>1<\/mn><mo>+<\/mo><mi>i<\/mi><msqrt>\n<mn>7<\/mn>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>i<\/mi><msqrt>\n<mn>7<\/mn>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mrow>\n<mo>\u2212<\/mo><mn>1<\/mn><mo>\u2212<\/mo><mi>i<\/mi><msqrt>\n<mn>7<\/mn>\n<\/msqrt>&nbsp;\n<\/mrow>\n<mn>2<\/mn>\n<\/mfrac>&nbsp;\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-id2370081\" 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_05_08\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2933152\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id2933153\">Solve the quadratic equation using the quadratic formula:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>9<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>9<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>3<\/mn><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>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2932184\" data-depth=\"1\">\n<h2 data-type=\"title\">The Discriminant<\/h2>\n<p id=\"fs-id1965271\">The <span id=\"term-00013\" class=\"no-emphasis\" data-type=\"term\">quadratic formula<\/span> not only generates the solutions to a quadratic equation, it tells us about the nature of the solutions when we consider the <span id=\"term-00014\" class=\"no-emphasis\" data-type=\"term\">discriminant<\/span>, or the expression under the radical,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>.<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>The discriminant tells us whether the solutions are real numbers or complex numbers, and how many solutions of each type to expect. <a class=\"autogenerated-content\" href=\"2-5-quadratic-equations#Table_02_05_01\">Table 1<\/a> relates the value of the discriminant to the solutions of a quadratic equation.<\/p>\n<div id=\"Table_02_05_01\" class=\"os-table\">\n<table data-id=\"Table_02_05_01\">\n<thead>\n<tr>\n<th scope=\"col\" data-align=\"center\">Value of Discriminant<\/th>\n<th scope=\"col\" data-align=\"center\">Results<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<td data-align=\"center\">One rational solution (double solution)<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>perfect square<\/td>\n<td data-align=\"center\">Two rational solutions<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>&gt;<\/mo><mn>0<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>not a perfect square<\/td>\n<td data-align=\"center\">Two irrational solutions<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\"><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>&lt;<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>&lt;<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/td>\n<td data-align=\"center\">Two complex solutions<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Table <\/span><br \/>\n<span class=\"os-number\">1<\/span><\/div>\n<\/div>\n<div id=\"fs-id2495380\" 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=\"\">The Discriminant<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"os-note-body\">\n<p id=\"fs-id1150951\">For<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>, where<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>b<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>b<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>, and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>c<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>c<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>are real numbers, the <span id=\"term-00015\" data-type=\"term\">discriminant<\/span> is the expression under the radical in the quadratic formula:<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>.<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>It tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"Example_02_05_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-id1340700\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1340702\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Using the Discriminant to Find the Nature of the Solutions to a Quadratic Equation<\/h3>\n<p id=\"fs-id2515703\">Use the discriminant to find the nature of the solutions to the following quadratic equations:<\/p>\n<ol id=\"fs-id2515706\" class=\"circled\" type=\"1\">\n<li><span class=\"token\">\u24d0<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li><span class=\"token\">\u24d1<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>8<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>14<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>8<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>14<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li><span class=\"token\">\u24d2<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/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>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<li><span class=\"token\">\u24d3<\/span><br \/>\n<math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>10<\/mn><mi>x<\/mi><mo>+<\/mo><mn>15<\/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>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>10<\/mn><mi>x<\/mi><mo>+<\/mo><mn>15<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<details id=\"fs-id2508928\" 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-id2508930\">Calculate the discriminant<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><\/mrow><\/annotation-xml><\/semantics><\/math>for each equation and state the expected type of solutions.<\/p>\n<ol id=\"fs-id1385650\" class=\"circled\" type=\"1\">\n<li><span class=\"token\">\u24d0<\/span>\n<p id=\"fs-id1699199\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>4<\/mn><mo>=<\/mo><mn>0<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p id=\"fs-id2390532\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>=<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mn>4<\/mn>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mrow><mo>(<\/mo>\n<mn>1<\/mn>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mn>4<\/mn>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>0.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>=<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mn>4<\/mn>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mrow><mo>(<\/mo>\n<mn>1<\/mn>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mn>4<\/mn>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>0.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>There will be one rational double solution.<\/li>\n<li><span class=\"token\">\u24d1<\/span>\n<p id=\"fs-id2933880\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>8<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>14<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>8<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>14<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><mo>=<\/mo><mn>0<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p id=\"fs-id2422396\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>=<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>14<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mrow><mo>(<\/mo>\n<mn>8<\/mn>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mn>3<\/mn>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>100.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>=<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>14<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mrow><mo>(<\/mo>\n<mn>8<\/mn>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mn>3<\/mn>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>100.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>As<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>100<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>100<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is a perfect square, there will be two rational solutions.<\/li>\n<li><span class=\"token\">\u24d2<\/span>\n<p id=\"fs-id1690680\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/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>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p id=\"fs-id1804761\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>=<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22125<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mrow><mo>(<\/mo>\n<mn>3<\/mn>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22122<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>49.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>=<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22125<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mrow><mo>(<\/mo>\n<mn>3<\/mn>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mn>\u22122<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>49.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>As<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>49<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>49<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is a perfect square, there will be two rational solutions.<\/li>\n<li><span class=\"token\">\u24d3<\/span>\n<p id=\"fs-id2016618\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mn>\u221210<\/mn><mi>x<\/mi><mo>+<\/mo><mn>15<\/mn><mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mn>\u221210<\/mn><mi>x<\/mi><mo>+<\/mo><mn>15<\/mn><mo>=<\/mo><mn>0<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p id=\"fs-id2905584\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>=<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u221210<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mrow><mo>(<\/mo>\n<mn>3<\/mn>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mn>15<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>\u221280.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>=<\/mo><msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>\u221210<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mrow><mo>(<\/mo>\n<mn>3<\/mn>\n<mo>)<\/mo><\/mrow><mrow><mo>(<\/mo>\n<mrow>\n<mn>15<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><mo>=<\/mo><mn>\u221280.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>There will be two complex solutions.<\/li>\n<\/ol>\n<\/div>\n<\/section>\n<\/details>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2443208\" data-depth=\"1\">\n<h2 data-type=\"title\">Using the Pythagorean Theorem<\/h2>\n<p id=\"fs-id2454436\">One of the most famous formulas in mathematics is the <span id=\"term-00016\" data-type=\"term\">Pythagorean Theorem<\/span>. It is based on a right triangle, and states the relationship among the lengths of the sides as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>c<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>c<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>where<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>b<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>b<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>refer to the legs of a right triangle adjacent to the<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>90\u00b0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>90\u00b0<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>angle, and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>c<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>c<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>refers to the hypotenuse. It has immeasurable uses in architecture, engineering, the sciences, geometry, trigonometry, and algebra, and in everyday applications.<\/p>\n<p id=\"fs-id2797112\">We use the Pythagorean Theorem to solve for the length of one side of a triangle when we have the lengths of the other two. Because each of the terms is squared in the theorem, when we are solving for a side of a triangle, we have a quadratic equation. We can use the methods for solving quadratic equations that we learned in this section to solve for the missing side.<\/p>\n<p id=\"fs-id2382116\">The Pythagorean Theorem is given as<\/p>\n<div id=\"fs-id2797115\" class=\"unnumbered\" data-type=\"equation\" data-label=\"\"><math display=\"block\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>c<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><msup>\n<mi>c<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math><\/div>\n<p id=\"fs-id1297273\">where<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>b<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>b<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>refer to the legs of a right triangle adjacent to the<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mrow>\n<mn>90<\/mn>\n<\/mrow>\n<mo>\u2218<\/mo>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mrow>\n<mn>90<\/mn>\n<\/mrow>\n<mo>\u2218<\/mo>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>angle, and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>c<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>c<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>refers to the hypotenuse, as shown in <a class=\"autogenerated-content\" href=\"2-5-quadratic-equations#Figure_02_05_004\">Figure 4<\/a>.<\/p>\n<div id=\"Figure_02_05_004\" class=\"os-figure\">\n<figure class=\"small\" data-id=\"Figure_02_05_004\"><span id=\"fs-id3040390\" data-type=\"media\" data-alt=\"Right triangle with the base labeled: a, the height labeled: b, and the hypotenuse labeled: c\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Right triangle with the base labeled: a, the height labeled: b, and the hypotenuse labeled: c\" width=\"248\" height=\"258\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/118170e707bfd06f5fedcf2dc304080f9308e31b\" \/><br \/>\n<\/span><\/figure>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span><br \/>\n<span class=\"os-number\">4<\/span><\/div>\n<\/div>\n<div id=\"Example_02_05_12\" 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\">12<\/span><\/h2>\n<\/header>\n<section>\n<div class=\"body\">\n<div id=\"fs-id2700450\" class=\"unnumbered\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2700452\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<h3 data-type=\"title\">Finding the Length of the Missing Side of a Right Triangle<\/h3>\n<p id=\"fs-id1887267\">Find the length of the missing side of the right triangle in <a class=\"autogenerated-content\" href=\"2-5-quadratic-equations#Figure_02_05_005\">Figure 5<\/a>.<\/p>\n<div id=\"Figure_02_05_005\" class=\"os-figure\">\n<figure class=\"small\" data-id=\"Figure_02_05_005\"><span id=\"fs-id3155491\" data-type=\"media\" data-alt=\"Right triangle with the base labeled: a, the height labeled: 4, and the hypotenuse labeled 12.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"Right triangle with the base labeled: a, the height labeled: 4, and the hypotenuse labeled 12.\" width=\"320\" height=\"183\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/aa457e668bedf9992e721efdd4bb275c8cc62b1c\" \/><br \/>\n<\/span><\/figure>\n<div class=\"os-caption-container\"><span class=\"os-title-label\">Figure <\/span><br \/>\n<span class=\"os-number\">5<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<details id=\"fs-id2308574\" 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-id2528218\">As we have measurements for side <em data-effect=\"italics\">b<\/em> and the hypotenuse, the missing side is <em data-effect=\"italics\">a.<\/em><\/p>\n<div id=\"fs-id2294487\" 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>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msup>\n<mi>c<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mn>2<\/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<mo stretchy=\"false\">(<\/mo><mn>12<\/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\" columnalign=\"right\">\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>16<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>144<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>128<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mi>a<\/mi>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>128<\/mn><\/mrow>\n<\/msqrt>\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\">\n<mrow>\n<mn>8<\/mn><msqrt>\n<mn>2<\/mn>\n<\/msqrt>\n<\/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<msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msup>\n<mi>c<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><msup>\n<mrow>\n<mo stretchy=\"false\">(<\/mo><mn>4<\/mn><mo stretchy=\"false\">)<\/mo><\/mrow>\n<mn>2<\/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<mo stretchy=\"false\">(<\/mo><mn>12<\/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\" columnalign=\"right\">\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>16<\/mn><\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>144<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mrow>\n<msup>\n<mi>a<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<\/mrow>\n<\/mtd>\n<mtd rowalign=\"center\"><mo>=<\/mo><\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<mn>128<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr rowalign=\"center\">\n<mtd rowalign=\"center\" columnalign=\"right\">\n<mi>a<\/mi>\n<\/mtd>\n<mtd rowalign=\"center\">\n<mo>=<\/mo>\n<\/mtd>\n<mtd rowalign=\"center\" columnalign=\"left\">\n<mrow>\n<msqrt>\n<mrow>\n<mn>128<\/mn><\/mrow>\n<\/msqrt>\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\">\n<mrow>\n<mn>8<\/mn><msqrt>\n<mn>2<\/mn>\n<\/msqrt>\n<\/mrow>\n<\/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-id1918460\" 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_05_09\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2381994\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<p id=\"fs-id2381996\">Use the Pythagorean Theorem to solve the right triangle problem: Leg <em data-effect=\"italics\">a <\/em>measures 4 units, leg <em data-effect=\"italics\">b <\/em>measures 3 units. Find the length of the hypotenuse.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1762359\" 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-id2666327\">Access these online resources for additional instruction and practice with quadratic equations.<\/p>\n<ul id=\"fs-id2666330\">\n<li><a href=\"http:\/\/openstax.org\/l\/quadreqfactor\" target=\"_blank\" rel=\"noopener nofollow\">Solving Quadratic Equations by Factoring<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/zeroprodprop\" target=\"_blank\" rel=\"noopener nofollow\">The Zero-Product Property<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/complthesqr\" target=\"_blank\" rel=\"noopener nofollow\">Completing the Square<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/quadrformrat\" target=\"_blank\" rel=\"noopener nofollow\">Quadratic Formula with Two Rational Solutions<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/leglengthtri\" target=\"_blank\" rel=\"noopener nofollow\">Length of a leg of a right triangle<\/a><\/li>\n<\/ul>\n<\/div>\n<\/section>\n<\/div>\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\">2.5 Section Exercises<\/span><\/h2>\n<section id=\"fs-id1685751\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id1685757\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id1752696\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1752698\" 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-id1752696-solution\">1<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1752699\">How do we recognize when an equation is quadratic?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1686335\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1686336\" 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-id1686337\">When we solve a quadratic equation, how many solutions should we always start out seeking? Explain why when solving a quadratic equation in the form<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>we may graph the equation<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi><mo>=<\/mo><mi>a<\/mi><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi><mo>=<\/mo><mi>a<\/mi><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and have no zeroes (<em data-effect=\"italics\">x<\/em>-intercepts).<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2440111\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3094972\" 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-id2440111-solution\">3<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3094973\">When we solve a quadratic equation by factoring, why do we move all terms to one side, having zero on the other side?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1540842\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1540843\" 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-id1540844\">In the quadratic formula, what is the name of the expression under the radical sign<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>b<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>4<\/mn><mi>a<\/mi><mi>c<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and how does it determine the number of and nature of our solutions?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2439837\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2439838\" 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-id2439837-solution\">5<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2439839\">Describe two scenarios where using the square root property to solve a quadratic equation would be the most efficient method.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id3039271\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<p id=\"fs-id3070461\">For the following exercises, solve the quadratic equation by factoring.<\/p>\n<div id=\"fs-id3070464\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3070466\" 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-id3070467\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>21<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><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 id=\"fs-id1227822\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1227823\" 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-id1227822-solution\">7<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1227824\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>9<\/mn><mi>x<\/mi><mo>+<\/mo><mn>18<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>9<\/mn><mi>x<\/mi><mo>+<\/mo><mn>18<\/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-id2029003\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2029004\" 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-id2029005\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>9<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>9<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/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-id1333066\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1333067\" 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-id1333066-solution\">9<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1333068\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>6<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>17<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/mn><mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>6<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>17<\/mn><mi>x<\/mi><mo>+<\/mo><mn>5<\/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-id2957156\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2957157\" 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-id2957158\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>4<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>12<\/mn><mi>x<\/mi><mo>+<\/mo><mn>8<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>4<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>12<\/mn><mi>x<\/mi><mo>+<\/mo><mn>8<\/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-id2708935\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2708936\" 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-id2708935-solution\">11<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2708937\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>75<\/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>2<\/mn>\n<\/msup>\n<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-id2933002\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2933003\" 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-id2933004\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>8<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>6<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>9<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>8<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>6<\/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<\/section>\n<\/div>\n<div id=\"fs-id1417836\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1417837\" 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-id1417836-solution\">13<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1417838\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>4<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>9<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>4<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>9<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1762775\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1762776\" 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-id1762777\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>14<\/mn><mi>x<\/mi><mo>=<\/mo><mn>36<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>14<\/mn><mi>x<\/mi><mo>=<\/mo><mn>36<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3207565\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3207566\" 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-id3207565-solution\">15<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3207567\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>5<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>30<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>5<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>30<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1929832\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1929833\" 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-id1395974\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>4<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>5<\/mn><mi>x<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>4<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>5<\/mn><mi>x<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1520388\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1520389\" 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-id1520388-solution\">17<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1520390\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>7<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>7<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><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-id2438766\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2438767\" 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-id2438768\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mfrac>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>9<\/mn>\n<mi>x<\/mi>\n<\/mfrac>\n<mo>=<\/mo><mn>2<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mfrac>\n<mi>x<\/mi>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>9<\/mn>\n<mi>x<\/mi>\n<\/mfrac>\n<mo>=<\/mo><mn>2<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2370838\">For the following exercises, solve the quadratic equation by using the square root property.<\/p>\n<div id=\"fs-id2370842\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2370843\" 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-id2370842-solution\">19<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1353332\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>36<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>36<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2049646\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2049648\" 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-id2049649\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>49<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>49<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1354953\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1354954\" 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-id1354953-solution\">21<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1846535\">\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<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>25<\/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<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>25<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2519565\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2519566\" 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-id2519567\">\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>=<\/mo><mn>7<\/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>=<\/mo><mn>7<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2977340\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2977341\" 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-id2977340-solution\">23<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2977342\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>9<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mrow>\n<mrow><mo>(<\/mo>\n<mrow>\n<mn>2<\/mn><mi>x<\/mi><mo>+<\/mo><mn>1<\/mn>\n<\/mrow>\n<mo>)<\/mo><\/mrow>\n<\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>9<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2643328\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2643329\" 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-id2643330\">\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>5<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\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>\u2212<\/mo><mn>5<\/mn><\/mrow>\n<mo>)<\/mo><\/mrow><\/mrow>\n<mn>2<\/mn>\n<\/msup>\n<mo>=<\/mo><mn>4<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id2432296\">For the following exercises, solve the quadratic equation by completing the square. Show each step.<\/p>\n<div id=\"fs-id2432300\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2432301\" 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-id2432300-solution\">25<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2432302\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<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>22<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<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>22<\/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-id3215750\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3215751\" 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-id3215752\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>8<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>8<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/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-id2980329\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2980330\" 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-id2980329-solution\">27<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2980331\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>6<\/mn><mi>x<\/mi><mo>=<\/mo><mn>13<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>6<\/mn><mi>x<\/mi><mo>=<\/mo><mn>13<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2020955\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2020956\" 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-id2020957\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mfrac>\n<mn>2<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\n<mi>x<\/mi><mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mn>3<\/mn>\n<\/mfrac>\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-id1840550\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1840551\" 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-id1840550-solution\">29<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1840552\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><mo>+<\/mo><mi>z<\/mi><mo>=<\/mo><mn>6<\/mn><msup>\n<mi>z<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><mo>+<\/mo><mi>z<\/mi><mo>=<\/mo><mn>6<\/mn><msup>\n<mi>z<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1552019\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1552020\" 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-id1552022\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>6<\/mn><msup>\n<mi>p<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>7<\/mn><mi>p<\/mi><mo>\u2212<\/mo><mn>20<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>6<\/mn><msup>\n<mi>p<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>7<\/mn><mi>p<\/mi><mo>\u2212<\/mo><mn>20<\/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-id1225471\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1225472\" 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-id1225471-solution\">31<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1225473\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/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<p id=\"fs-id1351674\">For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.<\/p>\n<div id=\"fs-id1351679\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1351680\" 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-id1351681\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>6<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/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-id2762702\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2762703\" 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-id2762702-solution\">33<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2762704\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/mn><mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>7<\/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-id2734018\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2734019\" 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-id2734020\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>8<\/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>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>8<\/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-id3142966\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3142967\" 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-id3142966-solution\">35<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3142968\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>9<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>30<\/mn><mi>x<\/mi><mo>+<\/mo><mn>25<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>9<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>30<\/mn><mi>x<\/mi><mo>+<\/mo><mn>25<\/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-id1357817\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1845325\" 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-id1845326\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/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-id1873764\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1873765\" 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-id1873764-solution\">37<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1873766\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>6<\/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<mn>6<\/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<p id=\"fs-id2947602\">For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state <em data-effect=\"italics\">No Real Solution<\/em>.<\/p>\n<div id=\"fs-id1883375\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1883376\" 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-id1883378\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/mn><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>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/mo><mn>3<\/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-id3070364\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3070365\" 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-id3070364-solution\">39<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3070366\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>=<\/mo><mn>4<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>=<\/mo><mn>4<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1846559\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1846560\" 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-id1846561\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>8<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/mn><mo>=<\/mo><mn>0<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>2<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>8<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>5<\/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-id1716230\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1716231\" 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-id1716230-solution\">41<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3274906\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>3<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/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>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>5<\/mn><mi>x<\/mi><mo>+<\/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-id1873745\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1873746\" 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-id1873747\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/mn><mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>4<\/mn><mi>x<\/mi><mo>+<\/mo><mn>2<\/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-id2919049\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2919050\" 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-id2919049-solution\">43<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2919051\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>4<\/mn><mo>+<\/mo><mfrac>\n<mn>1<\/mn>\n<mi>x<\/mi>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mn>0<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>4<\/mn><mo>+<\/mo><mfrac>\n<mn>1<\/mn>\n<mi>x<\/mi>\n<\/mfrac>\n<mo>\u2212<\/mo><mfrac>\n<mn>1<\/mn>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mfrac>\n<mo>=<\/mo><mn>0<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1421999\" data-depth=\"2\">\n<h3 data-type=\"title\">Technology<\/h3>\n<p id=\"fs-id1846578\">For the following exercises, enter the expressions into your graphing utility and find the zeroes to the equation (the <em data-effect=\"italics\">x<\/em>-intercepts) by using <strong>2<sup>nd<\/sup> CALC 2:zero<\/strong>. Recall finding zeroes will ask left bound (move your cursor to the left of the zero,enter), then right bound (move your cursor to the right of the zero,enter), then guess (move your cursor between the bounds near the zero, enter). Round your answers to the nearest thousandth.<\/p>\n<div id=\"fs-id1846592\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1846593\" 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-id1846594\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msub>\n<mtext>Y<\/mtext>\n<mn>1<\/mn>\n<\/msub>\n<mo>=<\/mo><mn>4<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msub>\n<mtext>Y<\/mtext>\n<mn>1<\/mn>\n<\/msub>\n<mo>=<\/mo><mn>4<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>3<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>2<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1204326\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1204327\" 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-id1204326-solution\">45<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1204328\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msub>\n<mtext>Y<\/mtext>\n<mn>1<\/mn>\n<\/msub>\n<mo>=<\/mo><mn>\u22123<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>8<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msub>\n<mtext>Y<\/mtext>\n<mn>1<\/mn>\n<\/msub>\n<mo>=<\/mo><mn>\u22123<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>8<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2506957\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2506958\" 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-id2506959\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msub>\n<mtext>Y<\/mtext>\n<mn>1<\/mn>\n<\/msub>\n<mo>=<\/mo><mn>0.5<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msub>\n<mtext>Y<\/mtext>\n<mn>1<\/mn>\n<\/msub>\n<mo>=<\/mo><mn>0.5<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\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-id1442695\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1442697\" 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-id1442695-solution\">47<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1442698\">To solve the quadratic equation<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><mo>=<\/mo><mn>4<\/mn><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><mo>=<\/mo><mn>4<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>we can graph these two equations<\/p>\n<p id=\"fs-id2442829\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msub>\n<mtext>Y<\/mtext>\n<mn>1<\/mn>\n<\/msub>\n<mo>=<\/mo><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msub>\n<mtext>Y<\/mtext>\n<mn>2<\/mn>\n<\/msub>\n<mo>=<\/mo><mn>4<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msub>\n<mtext>Y<\/mtext>\n<mn>1<\/mn>\n<\/msub>\n<mo>=<\/mo><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>5<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>7<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msub>\n<mtext>Y<\/mtext>\n<mn>2<\/mn>\n<\/msub>\n<mo>=<\/mo><mn>4<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p id=\"fs-id1240069\">and find the points of intersection. Recall 2<sup>nd<\/sup> CALC 5:intersection. Do this and find the solutions to the nearest tenth.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2697512\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2697513\" 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-id2495814\">To solve the quadratic equation<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>0.3<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mo>=<\/mo><mn>2<\/mn><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>0.3<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><mo>=<\/mo><mn>2<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>we can graph these two equations<\/p>\n<p id=\"fs-id2413227\">\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msub>\n<mtext>Y<\/mtext>\n<mn>1<\/mn>\n<\/msub>\n<mo>=<\/mo><mn>0.3<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msub>\n<mtext>Y<\/mtext>\n<mn>2<\/mn>\n<\/msub>\n<mo>=<\/mo><mn>2<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<mtable columnalign=\"left\">\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msub>\n<mtext>Y<\/mtext>\n<mn>1<\/mn>\n<\/msub>\n<mo>=<\/mo><mn>0.3<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>2<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>4<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>\n<mtr columnalign=\"left\">\n<mtd columnalign=\"left\">\n<mrow>\n<msub>\n<mtext>Y<\/mtext>\n<mn>2<\/mn>\n<\/msub>\n<mo>=<\/mo><mn>2<\/mn><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow>\n<\/mtd>\n<\/mtr>&nbsp;\n<\/mtable><\/mrow><\/annotation-xml><\/semantics><\/math>&nbsp;<\/p>\n<p id=\"fs-id2906629\">and find the points of intersection. Recall 2<sup>nd<\/sup> CALC 5:intersection. Do this and find the solutions to the nearest tenth.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id2498853\" data-depth=\"2\">\n<h3 data-type=\"title\">Extensions<\/h3>\n<div id=\"fs-id2498858\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2498859\" 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-id2498858-solution\">49<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2498860\">Beginning with the general form of a quadratic equation,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>a<\/mi><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>a<\/mi><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mi>b<\/mi><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>=<\/mo><mn>0<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>solve for <em data-effect=\"italics\">x<\/em> by using the completing the square method, thus deriving the quadratic formula.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2428496\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2428497\" data-type=\"problem\"><span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2428498\">Show that the sum of the two solutions to the quadratic equation is<\/p>\n<p><math display=\"inline\"><semantics><mrow><mo>\u2212<\/mo>\n<mfrac>\n<mrow>\n<mi>b<\/mi>\n<\/mrow>\n<mi>a<\/mi>\n<\/mfrac>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mo>\u2212<\/mo><mfrac>\n<mrow>\n<mi>b<\/mi>\n<\/mrow>\n<mi>a<\/mi>\n<\/mfrac><\/annotation-xml><\/semantics><\/math>.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2026640\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2021216\" 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-id2026640-solution\">51<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2021217\">A person has a garden that has a length 10 feet longer than the width. Set up a quadratic equation to find the dimensions of the garden if its area is 119 ft.<sup>2<\/sup>. Solve the quadratic equation to find the length and width.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2644744\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2644745\" data-type=\"problem\"><span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2720880\">Abercrombie and Fitch stock had a price given as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>P<\/mi><mo>=<\/mo><mn>0.2<\/mn><msup>\n<mi>t<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>5.6<\/mn><mi>t<\/mi><mo>+<\/mo><mn>50.2<\/mn><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>P<\/mi><mo>=<\/mo><mn>0.2<\/mn><msup>\n<mi>t<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>5.6<\/mn><mi>t<\/mi><mo>+<\/mo><mn>50.2<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>where<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>t<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>t<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is the time in months from 1999 to 2001. (<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>t<\/mi><mo>=<\/mo><mn>1<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>t<\/mi><mo>=<\/mo><mn>1<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is January 1999). Find the two months in which the price of the stock was $30.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id3113026\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id3113027\" 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-id3113026-solution\">53<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id3113028\">Suppose that an equation is given<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>p<\/mi><mo>=<\/mo><mn>\u22122<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>280<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1000<\/mn><mo>,<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>p<\/mi><mo>=<\/mo><mn>\u22122<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>280<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>1000<\/mn><mo>,<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>where<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>represents the number of items sold at an auction and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>p<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>p<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is the profit made by the business that ran the auction. How many items sold would make this profit a maximum? Solve this by graphing the expression in your graphing utility and finding the maximum using 2<sup>nd<\/sup> CALC maximum. To obtain a good window for the curve, set<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>x<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>x<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>[0,200] and<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>y<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>y<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>[0,10000].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1894925\" data-depth=\"2\">\n<h3 data-type=\"title\">Real-World Applications<\/h3>\n<div id=\"fs-id1894930\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2644764\" data-type=\"problem\"><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2644765\">A formula for the normal systolic blood pressure for a man age<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>A<\/mi><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>A<\/mi><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>measured in mmHg, is given as<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>P<\/mi><mo>=<\/mo><mn>0.006<\/mn><msup>\n<mi>A<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>0.02<\/mn><mi>A<\/mi><mo>+<\/mo><mn>120.<\/mn><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>P<\/mi><mo>=<\/mo><mn>0.006<\/mn><msup>\n<mi>A<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>\u2212<\/mo><mn>0.02<\/mn><mi>A<\/mi><mo>+<\/mo><mn>120.<\/mn><\/mrow><\/annotation-xml><\/semantics><\/math>Find the age to the nearest year of a man whose normal blood pressure measures 125 mmHg.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1700644\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1700645\" 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-id1700644-solution\">55<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id1700646\">The cost function for a certain company is<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>C<\/mi><mo>=<\/mo><mn>60<\/mn><mi>x<\/mi><mo>+<\/mo><mn>300<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>C<\/mi><mo>=<\/mo><mn>60<\/mn><mi>x<\/mi><mo>+<\/mo><mn>300<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>and the revenue is given by<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>R<\/mi><mo>=<\/mo><mn>100<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>0.5<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>.<\/mo><\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>R<\/mi><mo>=<\/mo><mn>100<\/mn><mi>x<\/mi><mo>\u2212<\/mo><mn>0.5<\/mn><msup>\n<mi>x<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>.<\/mo><\/mrow><\/annotation-xml><\/semantics><\/math>Recall that profit is revenue minus cost. Set up a quadratic equation and find two values of <em data-effect=\"italics\">x<\/em> (production level) that will create a profit of $300.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2454082\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2454083\" data-type=\"problem\"><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2454084\">A falling object travels a distance given by the formula<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>d<\/mi><mo>=<\/mo><mn>5<\/mn><mi>t<\/mi><mo>+<\/mo><mn>16<\/mn><msup>\n<mi>t<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>d<\/mi><mo>=<\/mo><mn>5<\/mn><mi>t<\/mi><mo>+<\/mo><mn>16<\/mn><msup>\n<mi>t<\/mi>\n<mn>2<\/mn>\n<\/msup>&nbsp;\n<\/mrow><\/annotation-xml><\/semantics><\/math>ft, where<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>t<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>t<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is measured in seconds. How long will it take for the object to travel 74 ft?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2413468\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2413469\" 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-id2413468-solution\">57<\/a><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2413470\">A vacant lot is being converted into a community garden. The garden and the walkway around its perimeter have an area of 378 ft<sup>2<\/sup>. Find the width of the walkway if the garden is 12 ft. wide by 15 ft. long.<\/p>\n<p><span id=\"fs-id2413475\" data-type=\"media\" data-alt=\"A rectangle inside of a larger rectangle. The smaller rectangle has the length labeled: 15 feet and the width labeled: 12 feet. The distance between the two rectangles is labeled as x on all four sides.\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/openstax.org\/books\/college-algebra-2e\/pages\/src#fixme\" alt=\"A rectangle inside of a larger rectangle. The smaller rectangle has the length labeled: 15 feet and the width labeled: 12 feet. The distance between the two rectangles is labeled as x on all four sides.\" width=\"378\" height=\"300\" data-media-type=\"image\/jpg\" data-lazy-src=\"\/apps\/archive\/20250226.165223\/resources\/884eec738c4dd93b9df83bfe221319139f5376fe\" \/><br \/>\n<\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id2385543\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id2385544\" data-type=\"problem\"><span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span><\/p>\n<div class=\"os-problem-container\">\n<p id=\"fs-id2385545\">An epidemiological study of the spread of a certain influenza strain that hit a small school population found that the total number of students,<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>P<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>P<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>, who contracted the flu<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>t<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>t<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>days after it broke out is given by the model<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>P<\/mi><mo>=<\/mo><mo>\u2212<\/mo><msup>\n<mi>t<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>13<\/mn><mi>t<\/mi><mo>+<\/mo><mn>130<\/mn><mo>,<\/mo>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>P<\/mi><mo>=<\/mo><mo>\u2212<\/mo><msup>\n<mi>t<\/mi>\n<mn>2<\/mn>\n<\/msup>\n<mo>+<\/mo><mn>13<\/mn><mi>t<\/mi><mo>+<\/mo><mn>130<\/mn><mo>,<\/mo>\n<\/mrow><\/annotation-xml><\/semantics><\/math>where<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mn>1<\/mn><mo>\u2264<\/mo><mi>t<\/mi><mo>\u2264<\/mo><mn>6.<\/mn>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mn>1<\/mn><mo>\u2264<\/mo><mi>t<\/mi><mo>\u2264<\/mo><mn>6.<\/mn>\n<\/mrow><\/annotation-xml><\/semantics><\/math>Find the day that 160 students had the flu. Recall that the restriction on<\/p>\n<p><math display=\"inline\"><semantics><mrow>\n<mrow>\n<mi>t<\/mi>\n<\/mrow>\n<\/mrow><annotation-xml encoding=\"MathML-Content\"><mrow>\n<mi>t<\/mi>\n<\/mrow><\/annotation-xml><\/semantics><\/math>is at most 6.<\/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":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-111","chapter","type-chapter","status-publish","hentry"],"part":51,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/111","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\/111\/revisions"}],"predecessor-version":[{"id":322,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/111\/revisions\/322"}],"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\/111\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=111"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=111"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=111"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=111"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}