{"id":109,"date":"2023-06-21T13:22:34","date_gmt":"2023-06-21T13:22:34","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-equation-solving-techniques\/"},"modified":"2023-06-21T13:22:34","modified_gmt":"2023-06-21T13:22:34","slug":"summary-equation-solving-techniques","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-equation-solving-techniques\/","title":{"raw":"Summary: Equation - Solving Techniques","rendered":"Summary: Equation &#8211; Solving Techniques"},"content":{"raw":"\n\n<h2>Key Concepts<\/h2>\n<ul>\n \t<li>Rational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve a radical equation, both sides of the equation are raised to a power that will render the exponent on the variable equal to 1.<\/li>\n \t<li>Factoring extends to higher-order polynomials when it involves factoring out the GCF or factoring by grouping.<\/li>\n \t<li>We can solve radical equations by isolating the radical and raising both sides of the equation to a power that matches the index.<\/li>\n \t<li>To solve absolute value equations, we need to write two equations, one for the positive value and one for the negative value.<\/li>\n \t<li>Equations in quadratic form are easy to spot, as the exponent on the first term is double the exponent on the second term and the third term is a constant. We may also see a binomial in place of the single variable. We use substitution to solve.<\/li>\n \t<li>Solving a rational equation may also lead to a quadratic equation or an equation in quadratic form.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n \t<dt><strong>absolute value equation<\/strong><\/dt>\n \t<dd id=\"fs-id1165137644990\">an equation in which the variable appears in absolute value bars, typically with two solutions, one accounting for the positive expression and one for the negative expression<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134297646\" class=\"definition\">\n \t<dt><strong>equations in quadratic form<\/strong><\/dt>\n \t<dd id=\"fs-id1165135486042\">equations with a power other than 2 but with a middle term with an exponent that is one-half the exponent of the leading term<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n \t<dt><strong>extraneous solutions<\/strong><\/dt>\n \t<dd id=\"fs-id1165137644990\">any solutions obtained that are not valid in the original equation<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134297646\" class=\"definition\">\n \t<dt><strong>polynomial equation<\/strong><\/dt>\n \t<dd id=\"fs-id1165135486042\">an equation containing a string of terms including numerical coefficients and variables raised to whole-number exponents<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n \t<dt><strong>radical equation<\/strong><\/dt>\n \t<dd id=\"fs-id1165137644990\">an equation containing at least one radical term where the variable is part of the radicand<\/dd>\n<\/dl>\n\n","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li>Rational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve a radical equation, both sides of the equation are raised to a power that will render the exponent on the variable equal to 1.<\/li>\n<li>Factoring extends to higher-order polynomials when it involves factoring out the GCF or factoring by grouping.<\/li>\n<li>We can solve radical equations by isolating the radical and raising both sides of the equation to a power that matches the index.<\/li>\n<li>To solve absolute value equations, we need to write two equations, one for the positive value and one for the negative value.<\/li>\n<li>Equations in quadratic form are easy to spot, as the exponent on the first term is double the exponent on the second term and the third term is a constant. We may also see a binomial in place of the single variable. We use substitution to solve.<\/li>\n<li>Solving a rational equation may also lead to a quadratic equation or an equation in quadratic form.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n<dt><strong>absolute value equation<\/strong><\/dt>\n<dd id=\"fs-id1165137644990\">an equation in which the variable appears in absolute value bars, typically with two solutions, one accounting for the positive expression and one for the negative expression<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134297646\" class=\"definition\">\n<dt><strong>equations in quadratic form<\/strong><\/dt>\n<dd id=\"fs-id1165135486042\">equations with a power other than 2 but with a middle term with an exponent that is one-half the exponent of the leading term<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n<dt><strong>extraneous solutions<\/strong><\/dt>\n<dd id=\"fs-id1165137644990\">any solutions obtained that are not valid in the original equation<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134297646\" class=\"definition\">\n<dt><strong>polynomial equation<\/strong><\/dt>\n<dd id=\"fs-id1165135486042\">an equation containing a string of terms including numerical coefficients and variables raised to whole-number exponents<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137644987\" class=\"definition\">\n<dt><strong>radical equation<\/strong><\/dt>\n<dd id=\"fs-id1165137644990\">an equation containing at least one radical term where the variable is part of the radicand<\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-109\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: OpenStax College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":395986,"menu_order":24,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"College Algebra\",\"author\":\"OpenStax College 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