{"id":46,"date":"2019-08-20T17:01:35","date_gmt":"2019-08-20T21:01:35","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/other-types-of-equations\/"},"modified":"2022-06-01T10:39:20","modified_gmt":"2022-06-01T14:39:20","slug":"other-types-of-equations","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/other-types-of-equations\/","title":{"raw":"Other Types of Equations","rendered":"Other Types of Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section you will:\n<ul>\n \t<li>Solve equations involving rational exponents.<\/li>\n \t<li>Solve equations using factoring.<\/li>\n \t<li>Solve radical equations.<\/li>\n \t<li>Solve absolute value equations.<\/li>\n \t<li>Solve other types of equations.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1519532\">We have solved linear equations, rational equations, and quadratic equations using several methods. However, there are many other types of equations, and we will investigate a few more types in this section. We will look at equations involving rational exponents, polynomial equations, radical equations, absolute value equations, equations in quadratic form, and some rational equations that can be transformed into quadratics. Solving any equation, however, employs the same basic algebraic rules. We will learn some new techniques as they apply to certain equations, but the algebra never changes.<\/p>\n\n<div id=\"fs-id2485804\" class=\"bc-section section\">\n<h3>Solving Equations Involving Rational Exponents<\/h3>\n<p id=\"fs-id1274976\">Rational exponents are exponents that are fractions, where the numerator is a power and the denominator is a root. For example,[latex]\\,{16}^{\\frac{1}{2}}\\,[\/latex]is another way of writing[latex]\\,\\sqrt{16};[\/latex][latex]{8}^{\\frac{1}{3}}\\,[\/latex]is another way of writing[latex]\\text{\u200b}\\,\\sqrt[3]{8}.\\,[\/latex]The ability to work with rational exponents is a useful skill, as it is highly applicable in calculus.<\/p>\n<p id=\"fs-id1723049\">We can solve equations in which a variable is raised to a rational exponent by raising both sides of the equation to the reciprocal of the exponent. The reason we raise the equation to the reciprocal of the exponent is because we want to eliminate the exponent on the variable term, and a number multiplied by its reciprocal equals 1. For example,[latex]\\,\\frac{2}{3}\\left(\\frac{3}{2}\\right)=1,[\/latex][latex]3\\left(\\frac{1}{3}\\right)=1,[\/latex]and so on.<\/p>\n\n<div id=\"fs-id1319605\" class=\"textbox key-takeaways\">\n<h3>Rational Exponents<\/h3>\n<p id=\"fs-id1532274\">A rational exponent indicates a power in the numerator and a root in the denominator. There are multiple ways of writing an expression, a variable, or a number with a rational exponent:<\/p>\n\n<div id=\"fs-id1512276\" class=\"unnumbered aligncenter\">[latex]{a}^{\\frac{m}{n}}={\\left({a}^{\\frac{1}{n}}\\right)}^{m}={\\left({a}^{m}\\right)}^{\\frac{1}{n}}=\\sqrt[n]{{a}^{m}}={\\left(\\sqrt[n]{a}\\right)}^{m}[\/latex]<\/div>\n<\/div>\n<div id=\"Example_02_06_01\" class=\"textbox examples\">\n<div id=\"fs-id1458449\">\n<div id=\"fs-id2592723\">\n<h3>Evaluating a Number Raised to a Rational Exponent<\/h3>\n<p id=\"fs-id1239738\">Evaluate[latex]\\,{8}^{\\frac{2}{3}}.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2422284\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2422284\"]\n<p id=\"fs-id2422284\">Whether we take the root first or the power first depends on the number. It is easy to find the cube root of 8, so rewrite[latex]\\,{8}^{\\frac{2}{3}}\\,[\/latex]as[latex]\\,{\\left({8}^{\\frac{1}{3}}\\right)}^{2}.[\/latex]<\/p>\n\n<div id=\"fs-id1227998\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill {\\left({8}^{\\frac{1}{3}}\\right)}^{2}&amp; =\\hfill &amp; {\\left(2\\right)}^{2}\\hfill \\\\ &amp; =&amp; 4\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1514351\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_06_01\">\n<div id=\"fs-id1336704\">\n<p id=\"fs-id2400251\">Evaluate[latex]\\,{64}^{-\\frac{1}{3}}.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2434546\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2434546\"]\n<p id=\"fs-id2434546\">[latex]\\frac{1}{4}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_02_06_02\" class=\"textbox examples\">\n<div id=\"fs-id2385021\">\n<div id=\"fs-id3264211\">\n<h3>Solve the Equation Including a Variable Raised to a Rational Exponent<\/h3>\n<p id=\"fs-id1198214\">Solve the equation in which a variable is raised to a rational exponent:[latex]\\,{x}^{\\frac{5}{4}}=32.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1569151\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1569151\"]\n<p id=\"fs-id1569151\">The way to remove the exponent on <em>x<\/em> is by raising both sides of the equation to a power that is the reciprocal of[latex]\\,\\frac{5}{4},[\/latex]which is[latex]\\,\\frac{4}{5}.[\/latex]<\/p>\n\n<div id=\"fs-id1277511\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cccc}\\hfill {x}^{\\frac{5}{4}}&amp; =&amp; 32\\hfill &amp; \\\\ \\hfill {\\left({x}^{\\frac{5}{4}}\\right)}^{\\frac{4}{5}}&amp; =&amp; {\\left(32\\right)}^{\\frac{4}{5}}\\hfill &amp; \\\\ \\hfill x&amp; =&amp; {\\left(2\\right)}^{4}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{The fifth root of 32 is 2.}\\hfill \\\\ &amp; =&amp; 16\\hfill &amp; \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1759744\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_06_02\">\n<div id=\"fs-id2293541\">\n<p id=\"fs-id1314798\">Solve the equation[latex]\\,{x}^{\\frac{3}{2}}=125.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"3268807\"]Show Solution[\/reveal-answer][hidden-answer a=\"3268807\"]\n[latex]25[\/latex][\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_02_06_03\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1335480\">\n<h3>Solving an Equation Involving Rational Exponents and Factoring<\/h3>\n<p id=\"fs-id1269017\">Solve[latex]\\,3{x}^{\\frac{3}{4}}={x}^{\\frac{1}{2}}.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"1522050\"]Show Solution[\/reveal-answer][hidden-answer a=\"1522050\"]\n\nThis equation involves rational exponents as well as factoring rational exponents. Let us take this one step at a time. First, put the variable terms on one side of the equal sign and set the equation equal to zero.\n<div id=\"fs-id1400272\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill 3{x}^{\\frac{3}{4}}-\\left({x}^{\\frac{1}{2}}\\right)&amp; =&amp; {x}^{\\frac{1}{2}}-\\left({x}^{\\frac{1}{2}}\\right)\\hfill \\\\ \\hfill 3{x}^{\\frac{3}{4}}-{x}^{\\frac{1}{2}}&amp; =&amp; 0\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1368636\">Now, it looks like we should factor the left side, but what do we factor out? We can always factor the term with the lowest exponent. Rewrite[latex]\\,{x}^{\\frac{1}{2}}\\,[\/latex]as[latex]\\,{x}^{\\frac{2}{4}}.\\,[\/latex]Then, factor out[latex]\\,{x}^{\\frac{2}{4}}\\,[\/latex]from both terms on the left.<\/p>\n\n<div id=\"fs-id1441452\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill 3{x}^{\\frac{3}{4}}-{x}^{\\frac{2}{4}}&amp; =&amp; 0\\hfill \\\\ \\hfill {x}^{\\frac{2}{4}}\\left(3{x}^{\\frac{1}{4}}-1\\right)&amp; =&amp; 0\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1933337\">Where did[latex]\\,{x}^{\\frac{1}{4}}\\,[\/latex]come from? Remember, when we multiply two numbers with the same base, we add the exponents. Therefore, if we multiply[latex]\\,{x}^{\\frac{2}{4}}\\,[\/latex]back in using the distributive property, we get the expression we had before the factoring, which is what should happen. We need an exponent such that when added to[latex]\\,\\frac{2}{4}\\,[\/latex]equals[latex]\\,\\frac{3}{4}.\\,[\/latex]Thus, the exponent on <em>x <\/em>in the parentheses is[latex]\\,\\frac{1}{4}.\\,[\/latex]<\/p>\n<p id=\"fs-id1891109\">Let us continue. Now we have two factors and can use the zero factor theorem.<\/p>\n\n<div id=\"fs-id1422353\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cccc}\\hfill {x}^{\\frac{2}{4}}\\left(3{x}^{\\frac{1}{4}}-1\\right)&amp; =&amp; 0\\hfill &amp; \\\\ \\hfill {x}^{\\frac{2}{4}}&amp; =&amp; 0\\hfill &amp; \\\\ \\hfill x&amp; =&amp; 0\\hfill &amp; \\\\ \\hfill 3{x}^{\\frac{1}{4}}-1&amp; =&amp; 0\\hfill &amp; \\\\ \\hfill 3{x}^{\\frac{1}{4}}&amp; =&amp; 1\\hfill &amp; \\\\ \\hfill {x}^{\\frac{1}{4}}&amp; =&amp; \\frac{1}{3}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Divide both sides by 3}.\\hfill \\\\ \\hfill {\\left({x}^{\\frac{1}{4}}\\right)}^{4}&amp; =&amp; {\\left(\\frac{1}{3}\\right)}^{4}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Raise both sides to the reciprocal of }\\frac{1}{4}.\\hfill \\\\ \\hfill x&amp; =&amp; \\frac{1}{81}\\hfill &amp; \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2957928\">The two solutions are [latex]\\,0[\/latex] and [latex]\\frac{1}{81}.[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1267637\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_06_03\">\n<div id=\"fs-id1411367\">\n<p id=\"fs-id1441991\">Solve:[latex]\\,{\\left(x+5\\right)}^{\\frac{3}{2}}=8.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1846504\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1846504\"]\n<p id=\"fs-id1846504\">[latex]\\left\\{-1\\right\\}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2945385\" class=\"bc-section section\">\n<h3>Solving Equations Using Factoring<\/h3>\n<p id=\"fs-id2521696\">We have used factoring to solve quadratic equations, but it is a technique that we can use with many types of polynomial equations, which are equations that contain a string of terms including numerical coefficients and variables. When we are faced with an equation containing polynomials of degree higher than 2, we can often solve them by factoring.<\/p>\n\n<div id=\"fs-id1786047\" class=\"textbox key-takeaways\">\n<h3>Polynomial Equations<\/h3>\n<p id=\"fs-id3231810\">A polynomial of degree <em>n <\/em>is an expression of the type<\/p>\n\n<div id=\"fs-id2905918\" class=\"unnumbered aligncenter\">[latex]{a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+\\,\\,\\,\\cdot \\,\\,\\cdot \\,\\,\\cdot \\,\\,+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/div>\n<p id=\"fs-id2444064\">where <em>n<\/em> is a positive integer and[latex]\\,{a}_{n},\\dots ,{a}_{0}\\,[\/latex]are real numbers and[latex]\\,{a}_{n}\\ne 0.[\/latex]<\/p>\n<p id=\"fs-id2931289\">Setting the polynomial equal to zero gives a polynomial equation. The total number of solutions (real and complex) to a polynomial equation is equal to the highest exponent <em>n<\/em>.<\/p>\n\n<\/div>\n<div id=\"Example_02_06_04\" class=\"textbox examples\">\n<div id=\"fs-id1996696\">\n<div id=\"fs-id2501149\">\n<h3>Solving a Polynomial by Factoring<\/h3>\n<p id=\"fs-id1391817\">Solve the polynomial by factoring:[latex]\\,5{x}^{4}=80{x}^{2}.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"1517483\"]Show Solution[\/reveal-answer][hidden-answer a=\"1517483\"]\n\nFirst, set the equation equal to zero. Then factor out what is common to both terms, the GCF.\n<div id=\"fs-id2502503\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill 5{x}^{4}-80{x}^{2}&amp; =&amp; 0\\hfill \\\\ \\hfill 5{x}^{2}\\left({x}^{2}-16\\right)&amp; =&amp; 0\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1924982\">Notice that we have the difference of squares in the factor[latex]\\,{x}^{2}-16,[\/latex]which we will continue to factor and obtain two solutions. The first term,[latex]\\,5{x}^{2},[\/latex]generates, technically, two solutions as the exponent is 2, but they are the same solution.<\/p>\n\n<div id=\"fs-id2824827\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill 5{x}^{2}&amp; =&amp; 0\\hfill \\\\ \\hfill x&amp; =&amp; 0\\hfill \\\\ \\hfill {x}^{2}-16&amp; =&amp; 0\\hfill \\\\ \\hfill \\left(x-4\\right)\\left(x+4\\right)&amp; =&amp; 0\\hfill \\\\ \\hfill x&amp; =&amp; 4\\hfill \\\\ \\hfill x&amp; =&amp; -4\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2638784\">The solutions are [latex]\\,0\\text{ (double solution),}[\/latex][latex]4,[\/latex] and [latex]\\,-4.[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<div id=\"fs-id1693367\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id768264\">We can see the solutions on the graph in <a class=\"autogenerated-content\" href=\"#Figure_02_06_001\">(Figure)<\/a>. The <em>x-<\/em>coordinates of the points where the graph crosses the <em>x-<\/em>axis are the solutions\u2014the <em>x-<\/em>intercepts. Notice on the graph that at the solution[latex]\\,0,[\/latex]the graph touches the <em>x-<\/em>axis and bounces back. It does not cross the <em>x-<\/em>axis. This is typical of double solutions.<\/p>\n\n<div id=\"Figure_02_06_001\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133613\/CNX_CAT_Figure_02_06_001.jpg\" alt=\"Coordinate plane with the x-axis ranging from negative 5 to 5 and the y-axis ranging from negative 400 to 500 in intervals of 100. The function five times x to the fourth power minus eighty x squared equals zero is graphed along with the points (negative 4,0), (0,0), and (4,0).\" width=\"487\" height=\"401\"> <strong>Figure 1.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n&nbsp;\n\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_06_04\">\n<div id=\"fs-id2496351\">\n<p id=\"fs-id3040346\">Solve by factoring:[latex]\\,12{x}^{4}=3{x}^{2}.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1244467\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1244467\"]\n<p id=\"fs-id1244467\">[latex]x=0,[\/latex][latex]x=\\frac{1}{2},[\/latex][latex]x=-\\frac{1}{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_02_06_05\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1007381\">\n<h3>Solve a Polynomial by Grouping<\/h3>\n<p id=\"fs-id2501656\">Solve a polynomial by grouping:[latex]\\,{x}^{3}+{x}^{2}-9x-9=0.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1422651\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1422651\"]\n<p id=\"fs-id1422651\">This polynomial consists of 4 terms, which we can solve by grouping. Grouping procedures require factoring the first two terms and then factoring the last two terms. If the factors in the parentheses are identical, we can continue the process and solve, unless more factoring is suggested.<\/p>\n\n<div id=\"fs-id2516826\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill {x}^{3}+{x}^{2}-9x-9&amp; =&amp; 0\\hfill \\\\ \\hfill {x}^{2}\\left(x+1\\right)-9\\left(x+1\\right)&amp; =&amp; 0\\hfill \\\\ \\hfill \\left({x}^{2}-9\\right)\\left(x+1\\right)&amp; =&amp; 0\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1408391\">The grouping process ends here, as we can factor[latex]\\,{x}^{2}-9\\,[\/latex]\nusing the difference of squares formula.<\/p>\n\n<div id=\"fs-id1402146\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\left({x}^{2}-9\\right)\\left(x+1\\right)&amp; =&amp; 0\\hfill \\\\ \\hfill \\left(x-3\\right)\\left(x+3\\right)\\left(x+1\\right)&amp; =&amp; 0\\hfill \\\\ \\hfill x&amp; =&amp; 3\\hfill \\\\ \\hfill x&amp; =&amp; -3\\hfill \\\\ \\hfill x&amp; =&amp; -1\\hfill \\end{array}[\/latex]<\/div>\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133616\/CNX_CAT_Figure_02_06_002.jpg\" alt=\"Coordinate plane with the x-axis ranging from negative 5 to 5 and the y-axis ranging from negative 30 to 20 in intervals of 5. The function x cubed plus x squared minus nine times x minus nine equals zero is graphed along with the points (negative 3,0), (negative 1,0), and (3,0).\" width=\"487\" height=\"438\"> <strong>Figure 2.<\/strong>[\/caption]\n<p id=\"fs-id2528732\">The solutions are [latex]3,[\/latex][latex]-3,[\/latex] and [latex]\\,-1.\\,[\/latex]Note that the highest exponent is 3 and we obtained 3 solutions. We can see the solutions, the <em>x-<\/em>intercepts, on the graph in <a class=\"autogenerated-content\" href=\"#Figure_02_06_002\">(Figure)<\/a>.<span id=\"fs-id2304259\"><\/span>[\/hidden-answer]<\/p>\n\n<\/div>\n<div id=\"fs-id2504109\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1243628\">We looked at solving quadratic equations by factoring when the leading coefficient is 1. When the leading coefficient is not 1, we solved by grouping. Grouping requires four terms, which we obtained by splitting the linear term of quadratic equations. We can also use grouping for some polynomials of degree higher than 2, as we saw here, since there were already four terms.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2506398\" class=\"bc-section section\">\n<h3>Solving Radical Equations<\/h3>\n<p id=\"fs-id1846458\"><strong>Radical equations<\/strong> are equations that contain variables in the <span class=\"no-emphasis\">radicand<\/span> (the expression under a radical symbol), such as<\/p>\n\n<div id=\"fs-id2781660\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\sqrt{3x+18}&amp; =&amp; x\\hfill \\\\ \\hfill \\sqrt{x+3}&amp; =&amp; x-3\\hfill \\\\ \\hfill \\sqrt{x+5}-\\sqrt{x-3}&amp; =&amp; 2\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1719179\">Radical equations may have one or more radical terms, and are solved by eliminating each radical, one at a time. We have to be careful when solving radical equations, as it is not unusual to find extraneous solutions, roots that are not, in fact, solutions to the equation. These solutions are not due to a mistake in the solving method, but result from the process of raising both sides of an equation to a power. However, checking each answer in the original equation will confirm the true solutions.<\/p>\n\n<div id=\"fs-id2506978\" class=\"textbox key-takeaways\">\n<h3>Radical Equations<\/h3>\n<p id=\"eip-id3052762\">An equation containing terms with a variable in the radicand is called a radical equation.<\/p>\n\n<\/div>\n<div id=\"fs-id2307719\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1517012\"><strong>Given a radical equation, solve it.<\/strong><\/p>\n\n<ol id=\"fs-id3143660\" type=\"1\">\n \t<li>Isolate the radical expression on one side of the equal sign. Put all remaining terms on the other side.<\/li>\n \t<li>If the radical is a square root, then square both sides of the equation. If it is a cube root, then raise both sides of the equation to the third power. In other words, for an <em>n<\/em>th root radical, raise both sides to the <em>n<\/em>th power. Doing so eliminates the radical symbol.<\/li>\n \t<li>Solve the remaining equation.<\/li>\n \t<li>If a radical term still remains, repeat steps 1\u20132.<\/li>\n \t<li>Confirm solutions by substituting them into the original equation.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_02_06_06\" class=\"textbox examples\">\n<div id=\"fs-id1222816\">\n<div id=\"fs-id1551039\">\n<h3>Solving an Equation with One Radical<\/h3>\n<p id=\"fs-id1520471\">Solve[latex]\\,\\sqrt{15-2x}=x.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1332305\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1332305\"]\n<p id=\"fs-id1332305\">The radical is already isolated on the left side of the equal side, so proceed to square both sides.<\/p>\n\n<div id=\"fs-id1919300\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\sqrt{15-2x}&amp; =&amp; x\\hfill \\\\ \\hfill {\\left(\\sqrt{15-2x}\\right)}^{2}&amp; =&amp; {\\left(x\\right)}^{2}\\hfill \\\\ \\hfill 15-2x&amp; =&amp; {x}^{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1895386\">We see that the remaining equation is a quadratic. Set it equal to zero and solve.<\/p>\n\n<div id=\"fs-id1905278\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill 0&amp; =&amp; {x}^{2}+2x-15\\hfill \\\\ &amp; =&amp; \\left(x+5\\right)\\left(x-3\\right)\\hfill \\\\ \\hfill x&amp; =&amp; -5\\hfill \\\\ \\hfill x&amp; =&amp; 3\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1257720\">The proposed solutions are [latex]-5\\,[\/latex] and [latex]3.\\,[\/latex]Let us check each solution back in the original equation. First, check[latex]\\,x=-5.[\/latex]<\/p>\n\n<div id=\"fs-id1463741\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\sqrt{15-2x}&amp; =&amp; x\\hfill \\\\ \\hfill \\sqrt{15-2\\left(-5\\right)}&amp; =&amp; -5\\hfill \\\\ \\hfill \\sqrt{25}&amp; =&amp; -5\\hfill \\\\ \\hfill 5&amp; \\ne &amp; -5\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2628354\">This is an extraneous solution. While no mistake was made solving the equation, we found a solution that does not satisfy the original equation.<\/p>\n<p id=\"fs-id1417225\">Check[latex]\\,x=3.[\/latex]<\/p>\n\n<div id=\"fs-id1515949\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\sqrt{15-2x}&amp; =&amp; x\\hfill \\\\ \\hfill \\sqrt{15-2\\left(3\\right)}&amp; =&amp; 3\\hfill \\\\ \\hfill \\sqrt{9}&amp; =&amp; 3\\hfill \\\\ \\hfill 3&amp; =&amp; 3\\hfill \\end{array}[\/latex]<\/div>\nThe solution is [latex]\\,3.[\/latex][\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1441693\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_06_05\">\n<div id=\"fs-id1583879\">\n<p id=\"fs-id1583880\">Solve the radical equation:[latex]\\,\\sqrt{x+3}=3x-1[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1535940\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1535940\"]\n<p id=\"fs-id1535940\">[latex]x=1;[\/latex]extraneous solution[latex]\\,x=-\\frac{2}{9}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_02_06_07\" class=\"textbox examples\">\n<div id=\"fs-id1720874\">\n<div id=\"fs-id2499586\">\n<h3>Solving a Radical Equation Containing Two Radicals<\/h3>\n<p id=\"fs-id1410869\">Solve[latex]\\,\\sqrt{2x+3}+\\sqrt{x-2}=4.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1929508\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1929508\"]\n<p id=\"fs-id1929508\">As this equation contains two radicals, we isolate one radical, eliminate it, and then isolate the second radical.<\/p>\n\n<div id=\"fs-id2268062\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cccc}\\hfill \\sqrt{2x+3}+\\sqrt{x-2}&amp; =&amp; 4\\hfill &amp; \\\\ \\hfill \\sqrt{2x+3}&amp; =&amp; 4-\\sqrt{x-2}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Subtract }\\sqrt{x-2}\\text{ from both sides}.\\hfill \\\\ \\hfill {\\left(\\sqrt{2x+3}\\right)}^{2}&amp; =&amp; {\\left(4-\\sqrt{x-2}\\right)}^{2}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Square both sides}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1548786\">Use the perfect square formula to expand the right side:[latex]\\,{\\left(a-b\\right)}^{2}={a}^{2}-2ab+{b}^{2}.[\/latex]<\/p>\n\n<div id=\"fs-id2506054\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cccc}\\hfill 2x+3&amp; =&amp; {\\left(4\\right)}^{2}-2\\left(4\\right)\\sqrt{x-2}+{\\left(\\sqrt{x-2}\\right)}^{2}\\hfill &amp; \\\\ \\hfill 2x+3&amp; =&amp; 16-8\\sqrt{x-2}+\\left(x-2\\right)\\hfill &amp; \\\\ \\hfill 2x+3&amp; =&amp; 14+x-8\\sqrt{x-2}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Combine like terms}.\\hfill \\\\ \\hfill x-11&amp; =&amp; -8\\sqrt{x-2}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Isolate the second radical}.\\hfill \\\\ \\hfill {\\left(x-11\\right)}^{2}&amp; =&amp; {\\left(-8\\sqrt{x-2}\\right)}^{2}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Square both sides}.\\hfill \\\\ \\hfill {x}^{2}-22x+121&amp; =&amp; 64\\left(x-2\\right)\\hfill &amp; \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1175462\">Now that both radicals have been eliminated, set the quadratic equal to zero and solve.<\/p>\n\n<div id=\"fs-id2946619\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cccc}\\hfill {x}^{2}-22x+121&amp; =&amp; 64x-128\\hfill &amp; \\\\ \\hfill {x}^{2}-86x+249&amp; =&amp; 0\\hfill &amp; \\\\ \\hfill \\left(x-3\\right)\\left(x-83\\right)&amp; =&amp; 0\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Factor and solve}.\\hfill \\\\ \\hfill x&amp; =&amp; 3\\hfill &amp; \\\\ \\hfill x&amp; =&amp; 83\\hfill &amp; \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1563543\">The proposed solutions are [latex]\\,3\\,[\/latex] and [latex]\\,83.\\,[\/latex]Check each solution in the original equation.<\/p>\n\n<div id=\"fs-id2655511\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\sqrt{2x+3}+\\sqrt{x-2}&amp; =&amp; 4\\hfill \\\\ \\hfill \\sqrt{2x+3}&amp; =&amp; 4-\\sqrt{x-2}\\hfill \\\\ \\hfill \\sqrt{2\\left(3\\right)+3}&amp; =&amp; 4-\\sqrt{\\left(3\\right)-2}\\hfill \\\\ \\hfill \\sqrt{9}&amp; =&amp; 4-\\sqrt{1}\\hfill \\\\ \\hfill 3&amp; =&amp; 3\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1840544\">One solution is [latex]\\,3.[\/latex]<\/p>\n<p id=\"fs-id1918601\">Check[latex]\\,x=83.[\/latex]<\/p>\n\n<div id=\"fs-id1941177\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\sqrt{2x+3}+\\sqrt{x-2}&amp; =&amp; 4\\hfill \\\\ \\hfill \\sqrt{2x+3}&amp; =&amp; 4-\\sqrt{x-2}\\hfill \\\\ \\hfill \\sqrt{2\\left(83\\right)+3}&amp; =&amp; 4-\\sqrt{\\left(83-2\\right)}\\hfill \\\\ \\hfill \\sqrt{169}&amp; =&amp; 4-\\sqrt{81}\\hfill \\\\ \\hfill 13&amp; \\ne &amp; -5\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2293780\">The only solution is [latex]\\,3.\\,[\/latex]We see that[latex]\\,x=83\\,[\/latex]is an extraneous solution.[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2430499\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_06_06\">\n<div id=\"fs-id2702970\">\n<p id=\"fs-id2429740\">Solve the equation with two radicals:[latex]\\,\\sqrt{3x+7}+\\sqrt{x+2}=1.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1951350\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1951350\"]\n<p id=\"fs-id1951350\">[latex]x=-2;[\/latex]extraneous solution[latex]\\,x=-1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id3240737\" class=\"bc-section section\">\n<h3>Solving an Absolute Value Equation<\/h3>\n<p id=\"fs-id1335626\">Next, we will learn how to solve an <span class=\"no-emphasis\">absolute value equation<\/span>. To solve an equation such as[latex]\\,|2x-6|=8,[\/latex]we notice that the absolute value will be equal to 8 if the quantity inside the absolute value bars is[latex]\\,8\\,[\/latex]or[latex]\\,-8.\\,[\/latex]This leads to two different equations we can solve independently.<\/p>\n\n<div id=\"fs-id2638694\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccccccc}\\hfill 2x-6&amp; =&amp; 8\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{or}\\phantom{\\rule{2em}{0ex}}&amp; \\hfill 2x-6&amp; =&amp; -8\\hfill \\\\ \\hfill 2x&amp; =&amp; 14&amp; &amp; \\hfill 2x&amp; =&amp; -2\\hfill \\\\ \\hfill x&amp; =&amp; 7\\hfill &amp; &amp; \\hfill x&amp; =&amp; -1\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1280820\">Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.<\/p>\n\n<div id=\"fs-id2918915\" class=\"textbox key-takeaways\">\n<h3>Absolute Value Equations<\/h3>\n<p id=\"fs-id1542481\">The absolute value of <em>x <\/em>is written as[latex]\\,|x|.\\,[\/latex]It has the following properties:<\/p>\n\n<div id=\"fs-id2434805\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{If }x\\ge 0,\\text{ then }|x|=x.\\hfill \\\\ \\text{If }x&lt;0,\\text{ then }|x|=-x.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2905392\">For real numbers[latex]\\,A\\,[\/latex]and[latex]\\,B,[\/latex]an equation of the form[latex]\\,|A|=B,[\/latex]with[latex]\\,B\\ge 0,[\/latex]will have solutions when[latex]\\,A=B\\,[\/latex]or[latex]\\,A=-B.\\,[\/latex]If[latex]\\,B&lt;0,[\/latex]the equation[latex]\\,|A|=B\\,[\/latex]has no solution.<\/p>\n<p id=\"fs-id1408140\">An absolute value equation in the form[latex]\\,|ax+b|=c\\,[\/latex]has the following properties:<\/p>\n\n<div id=\"fs-id2655133\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{If }c&lt;0,|ax+b|=c\\text{ has no solution}.\\hfill \\\\ \\text{If }c=0,|ax+b|=c\\text{ has one solution}.\\hfill \\\\ \\text{If }c&gt;0,|ax+b|=c\\text{ has two solutions}.\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id3038796\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1793504\"><strong>Given an absolute value equation, solve it.<\/strong><\/p>\n\n<ol id=\"fs-id2938958\" type=\"1\">\n \t<li>Isolate the absolute value expression on one side of the equal sign.<\/li>\n \t<li>If[latex]\\,c&gt;0,[\/latex]write and solve two equations:[latex]\\,ax+b=c\\,[\/latex]and[latex]\\,ax+b=-c.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_02_06_08\" class=\"textbox examples\">\n<div id=\"fs-id3105697\">\n<div id=\"fs-id1836188\">\n<h3>Solving Absolute Value Equations<\/h3>\n<p id=\"fs-id1836194\">Solve the following absolute value equations:<\/p>\n\n<ul id=\"fs-id2370595\">\n \t<li>(a) [latex]|6x+4|=8[\/latex]<\/li>\n \t<li>(b) [latex]|3x+4|=-9[\/latex]<\/li>\n \t<li>(c) [latex]|3x-5|-4=6[\/latex]<\/li>\n \t<li>(d) [latex]|-5x+10|=0[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"2683627\"]Show Solution[\/reveal-answer][hidden-answer a=\"2683627\"]\n<ul id=\"fs-id2683629\">\n \t<li>\n<p id=\"fs-id1832720\">(a) [latex]|6x+4|=8[\/latex]<\/p>\n<p id=\"fs-id2389886\">Write two equations and solve each:<\/p>\n\n<div id=\"eip-id1170402604810\" class=\"unnumbered\">[latex]\\begin{array}{ccccccc}\\hfill 6x+4&amp; =&amp; 8\\hfill &amp; \\phantom{\\rule{5em}{0ex}}&amp; \\hfill 6x+4&amp; =&amp; -8\\hfill \\\\ \\hfill 6x&amp; =&amp; 4\\hfill &amp; \\phantom{\\rule{5em}{0ex}}&amp; \\hfill 6x&amp; =&amp; -12\\hfill \\\\ \\hfill x&amp; =&amp; \\frac{2}{3}\\hfill &amp; \\phantom{\\rule{5em}{0ex}}&amp; \\hfill x&amp; =&amp; -2\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id3203361\">The two solutions are [latex]\\,\\frac{2}{3}[\/latex] and [latex]-2.[\/latex]<\/p>\n<\/li>\n \t<li>\n<p id=\"fs-id1879343\">(b) [latex]|3x+4|=-9[\/latex]<\/p>\n<p id=\"fs-id2293707\">There is no solution as an absolute value cannot be negative.<\/p>\n<\/li>\n \t<li>\n<p id=\"fs-id2888792\">(c) [latex]|3x-5|-4=6[\/latex]<\/p>\n<p id=\"fs-id2436076\">Isolate the absolute value expression and then write two equations.<\/p>\n\n<div id=\"fs-id3274908\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccccccccc}&amp; &amp; &amp; \\hfill |3x-5|-4&amp; =&amp; 6\\hfill &amp; &amp; &amp; \\\\ &amp; &amp; &amp; \\hfill |3x-5|&amp; =&amp; 10\\hfill &amp; &amp; &amp; \\\\ \\hfill 3x-5&amp; =&amp; 10\\hfill &amp; &amp; &amp; &amp; \\hfill 3x-5&amp; =&amp; -10\\hfill \\\\ \\hfill 3x&amp; =&amp; 15\\hfill &amp; &amp; &amp; &amp; \\hfill 3x&amp; =&amp; -5\\hfill \\\\ \\hfill x&amp; =&amp; 5\\hfill &amp; &amp; &amp; &amp; \\hfill x&amp; =&amp; -\\frac{5}{3}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2498828\">There are two solutions: [latex]\\,5,[\/latex] and [latex]-\\frac{5}{3}.[\/latex]<\/p>\n<\/li>\n \t<li>\n<p id=\"fs-id1340685\">(d) [latex]|-5x+10|=0[\/latex]<\/p>\nThe equation is set equal to zero, so we have to write only one equation.\n<div id=\"fs-id2431537\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill -5x+10&amp; =&amp; 0\\hfill \\\\ \\hfill -5x&amp; =&amp; -10\\hfill \\\\ \\hfill x&amp; =&amp; 2\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1351555\">There is one solution: [latex]\\,2.[\/latex][\/hidden-answer]<\/p>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1535632\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_06_07\">\n<div id=\"fs-id2515512\">\n<p id=\"fs-id2515513\">Solve the absolute value equation:[latex]|1-4x|+8=13.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2385860\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2385860\"]\n<p id=\"fs-id2385860\">[latex]x=-1,[\/latex][latex]x=\\frac{3}{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2653809\" class=\"bc-section section\">\n<h3>Solving Other Types of Equations<\/h3>\n<p id=\"fs-id3169548\">There are many other types of equations in addition to the ones we have discussed so far. We will see more of them throughout the text. Here, we will discuss equations that are in quadratic form, and rational equations that result in a quadratic.<\/p>\n\n<div id=\"fs-id1923869\" class=\"bc-section section\">\n<h4>Solving Equations in Quadratic Form<\/h4>\n<p id=\"fs-id1923873\"><strong>Equations in quadratic form <\/strong>are equations with three terms. The first term has a power other than 2. The middle term has an exponent that is one-half the exponent of the leading term. The third term is a constant. We can solve equations in this form as if they were quadratic. A few examples of these equations include[latex]\\,{x}^{4}-5{x}^{2}+4=0,{x}^{6}+7{x}^{3}-8=0,[\/latex]and[latex]\\,{x}^{\\frac{2}{3}}+4{x}^{\\frac{1}{3}}+2=0.\\,[\/latex]In each one, doubling the exponent of the middle term equals the exponent on the leading term. We can solve these equations by substituting a variable for the middle term.<\/p>\n\n<div id=\"fs-id1834422\" class=\"textbox key-takeaways\">\n<h3>Quadratic Form<\/h3>\n<p id=\"fs-id3094567\">If the exponent on the middle term is one-half of the exponent on the leading term, we have an equation in quadratic form, which we can solve as if it were a quadratic. We substitute a variable for the middle term to solve equations in quadratic form.<\/p>\n\n<\/div>\n<div id=\"fs-id1954745\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1385726\"><strong>Given an equation quadratic in form, solve it.<\/strong><\/p>\n\n<ol id=\"fs-id1385730\" type=\"1\">\n \t<li>Identify the exponent on the leading term and determine whether it is double the exponent on the middle term.<\/li>\n \t<li>If it is, substitute a variable, such as <em>u<\/em>, for the variable portion of the middle term.<\/li>\n \t<li>Rewrite the equation so that it takes on the standard form of a quadratic.<\/li>\n \t<li>Solve using one of the usual methods for solving a quadratic.<\/li>\n \t<li>Replace the substitution variable with the original term.<\/li>\n \t<li>Solve the remaining equation.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_02_06_09\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1357816\">\n<h3>Solving a Fourth-degree Equation in Quadratic Form<\/h3>\n<p id=\"fs-id2486573\">Solve this fourth-degree equation:[latex]\\,3{x}^{4}-2{x}^{2}-1=0.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1229229\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1229229\"]\n<p id=\"fs-id1229229\">This equation fits the main criteria, that the power on the leading term is double the power on the middle term. Next, we will make a substitution for the variable term in the middle. Let[latex]\\,u={x}^{2}.[\/latex]Rewrite the equation in <em>u<\/em>.<\/p>\n\n<div class=\"unnumbered\">[latex]3{u}^{2}-2u-1=0[\/latex]<\/div>\n<p id=\"fs-id2371956\">Now solve the quadratic.<\/p>\n\n<div id=\"fs-id2371960\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill 3{u}^{2}-2u-1&amp; =&amp; 0\\hfill \\\\ \\hfill \\left(3u+1\\right)\\left(u-1\\right)&amp; =&amp; 0\\hfill \\end{array}[\/latex]<\/div>\nSolve each factor and replace the original term for <em>u.<\/em>\n<div id=\"fs-id2797268\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill 3u+1&amp; =&amp; 0\\hfill \\\\ \\hfill 3u&amp; =&amp; -1\\hfill \\\\ \\hfill u&amp; =&amp; -\\frac{1}{3}\\hfill \\\\ \\hfill {x}^{2}&amp; =&amp; -\\frac{1}{3}\\hfill \\\\ \\hfill x&amp; =&amp; \u00b1i\\sqrt{\\frac{1}{3}}\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"fs-id1335933\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill u-1&amp; =&amp; 0\\hfill \\\\ \\hfill u&amp; =&amp; 1\\hfill \\\\ \\hfill {x}^{2}&amp; =&amp; 1\\hfill \\\\ \\hfill x&amp; =&amp; \u00b11\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2980376\">The solutions are [latex]\\,\u00b1i\\sqrt{\\frac{1}{3}}\\,[\/latex] and [latex]\\,\u00b11.[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1929412\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_06_08\">\n<div id=\"fs-id2514218\">\n<p id=\"fs-id2514219\">Solve using substitution:[latex]\\,{x}^{4}-8{x}^{2}-9=0.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1929399\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1929399\"]\n<p id=\"fs-id1929399\">[latex]x=-3,3,-i,i[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_02_06_10\" class=\"textbox examples\">\n<div id=\"fs-id2444359\">\n<div id=\"fs-id2444361\">\n<h3>Solving an Equation in Quadratic Form Containing a Binomial<\/h3>\n<p id=\"fs-id2029548\">Solve the equation in quadratic form:[latex]\\,{\\left(x+2\\right)}^{2}+11\\left(x+2\\right)-12=0.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2382012\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2382012\"]\n<p id=\"fs-id2382012\">This equation contains a binomial in place of the single variable. The tendency is to expand what is presented. However, recognizing that it fits the criteria for being in quadratic form makes all the difference in the solving process. First, make a substitution, letting[latex]\\,u=x+2.\\,[\/latex]Then rewrite the equation in <em>u.<\/em><\/p>\n\n<div id=\"fs-id1929605\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill {u}^{2}+11u-12&amp; =&amp; 0\\hfill \\\\ \\hfill \\left(u+12\\right)\\left(u-1\\right)&amp; =&amp; 0\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1691070\">Solve using the zero-factor property and then replace <em>u<\/em> with the original expression.<\/p>\n\n<div id=\"fs-id1516932\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill u+12&amp; =&amp; 0\\hfill \\\\ \\hfill u&amp; =&amp; -12\\hfill \\\\ \\hfill x+2&amp; =&amp; -12\\hfill \\\\ \\hfill x&amp; =&amp; -14\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1720759\">The second factor results in<\/p>\n\n<div id=\"fs-id2485990\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill u-1&amp; =&amp; 0\\hfill \\\\ \\hfill u&amp; =&amp; 1\\hfill \\\\ \\hfill x+2&amp; =&amp; 1\\hfill \\\\ \\hfill x&amp; =&amp; -1\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1943470\">We have two solutions: [latex]\\,-14,[\/latex] and [latex]-1.[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_06_09\">\n<div id=\"fs-id1952007\">\n<p id=\"fs-id1952008\">Solve:[latex]\\,{\\left(x-5\\right)}^{2}-4\\left(x-5\\right)-21=0.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1789429\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1789429\"]\n<p id=\"fs-id1789429\">[latex]x=2,x=12[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2893053\" class=\"bc-section section\">\n<h4>Solving Rational Equations Resulting in a Quadratic<\/h4>\n<p id=\"fs-id1679917\">Earlier, we solved rational equations. Sometimes, solving a rational equation results in a quadratic. When this happens, we continue the solution by simplifying the quadratic equation by one of the methods we have seen. It may turn out that there is no solution.<\/p>\n\n<div id=\"Example_02_06_11\" class=\"textbox examples\">\n<div id=\"fs-id1970290\">\n<div id=\"fs-id2508561\">\n<h3>Solving a Rational Equation Leading to a Quadratic<\/h3>\n<p id=\"fs-id2508566\">Solve the following rational equation:[latex]\\,\\frac{-4x}{x-1}+\\frac{4}{x+1}=\\frac{-8}{{x}^{2}-1}.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2797287\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2797287\"]\n<p id=\"fs-id2797287\">We want all denominators in factored form to find the LCD. Two of the denominators cannot be factored further. However,[latex]\\,{x}^{2}-1=\\left(x+1\\right)\\left(x-1\\right).\\,[\/latex]Then, the LCD is[latex]\\,\\left(x+1\\right)\\left(x-1\\right).\\,[\/latex]Next, we multiply the whole equation by the LCD.<\/p>\n\n<div id=\"fs-id2441856\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\left(x+1\\right)\\left(x-1\\right)\\left[\\frac{-4x}{x-1}+\\frac{4}{x+1}\\right]&amp; =&amp; \\left[\\frac{-8}{\\left(x+1\\right)\\left(x-1\\right)}\\right]\\left(x+1\\right)\\left(x-1\\right)\\hfill \\\\ \\hfill -4x\\left(x+1\\right)+4\\left(x-1\\right)&amp; =&amp; -8\\hfill \\\\ \\hfill -4{x}^{2}-4x+4x-4&amp; =&amp; -8\\hfill \\\\ \\hfill -4{x}^{2}+4&amp; =&amp; 0\\hfill \\\\ \\hfill -4\\left({x}^{2}-1\\right)&amp; =&amp; 0\\hfill \\\\ \\hfill -4\\left(x+1\\right)\\left(x-1\\right)&amp; =&amp; 0\\hfill \\\\ \\hfill x&amp; =&amp; -1\\hfill \\\\ \\hfill x&amp; =&amp; 1\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2980350\">In this case, either solution produces a zero in the denominator in the original equation. Thus, there is no solution.[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2906702\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_06_10\">\n<div id=\"fs-id1751154\">\n<p id=\"fs-id1751155\">Solve[latex]\\,\\frac{3x+2}{x-2}+\\frac{1}{x}=\\frac{-2}{{x}^{2}-2x}.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1831304\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1831304\"]\n<p id=\"fs-id1831304\">[latex]x=-1,[\/latex][latex]x=0[\/latex]is not a solution.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2665007\" class=\"precalculus media\">\n<p id=\"fs-id1822998\">Access these online resources for additional instruction and practice with different types of equations.<\/p>\n\n<ul id=\"fs-id1823002\">\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/rateqnosoln\">Rational Equation with no Solution<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/ratexprecpexp\">Solving equations with rational exponents using reciprocal powers<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/radeqsolvepart1\">Solving radical equations part 1 of 2<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/radeqsolvepart2\">Solving radical equations part 2 of 2<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2755534\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id3148376\">\n \t<li>Rational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve, both sides of the equation are raised to a power that will render the exponent on the variable equal to 1. See <a class=\"autogenerated-content\" href=\"#Example_02_06_01\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_02_06_02\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_02_06_03\">(Figure)<\/a>.<\/li>\n \t<li>Factoring extends to higher-order polynomials when it involves factoring out the GCF or factoring by grouping. See <a class=\"autogenerated-content\" href=\"#Example_02_06_04\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_02_06_05\">(Figure)<\/a>.<\/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. See <a class=\"autogenerated-content\" href=\"#Example_02_06_06\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_02_06_07\">(Figure)<\/a>.<\/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. See <a class=\"autogenerated-content\" href=\"#Example_02_06_08\">(Figure)<\/a>.<\/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. See <a class=\"autogenerated-content\" href=\"#Example_02_06_09\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_02_06_10\">(Figure)<\/a>.<\/li>\n \t<li>Solving a rational equation may also lead to a quadratic equation or an equation in quadratic form. See <a class=\"autogenerated-content\" href=\"#Example_02_06_11\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id3146179\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id3146185\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id2028956\">\n<div id=\"fs-id2028957\">\n<p id=\"fs-id2028958\">In a radical equation, what does it mean if a number is an extraneous solution?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2028962\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2028962\"]\n<p id=\"fs-id2028962\">This is not a solution to the radical equation, it is a value obtained from squaring both sides and thus changing the signs of an equation which has caused it not to be a solution in the original equation.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id3256935\">\n<div id=\"fs-id3256936\">\n<p id=\"fs-id3256937\">Explain why possible solutions <em>must<\/em> be checked in radical equations.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2501234\">\n<div id=\"fs-id2501235\">\n<p id=\"fs-id2501236\">Your friend tries to calculate the value[latex]\\,-{9}^{\\frac{3}{2}}[\/latex]and keeps getting an ERROR message. What mistake is he or she probably making?<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1690245\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1690245\"]\n<p id=\"fs-id1690245\">He or she is probably trying to enter negative 9, but taking the square root of[latex]\\,-9\\,[\/latex]is not a real number. The negative sign is in front of this, so your friend should be taking the square root of 9, cubing it, and then putting the negative sign in front, resulting in[latex]\\,-27.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1941430\">\n<div id=\"fs-id1941431\">\n<p id=\"fs-id1941432\">Explain why[latex]\\,|2x+5|=-7\\,[\/latex]has no solutions.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1354283\">\n<div id=\"fs-id1354284\">\n<p id=\"fs-id1354285\">Explain how to change a rational exponent into the correct radical expression.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id3018943\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id3018943\"]\n<p id=\"fs-id3018943\">A rational exponent is a fraction: the denominator of the fraction is the root or index number and the numerator is the power to which it is raised.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id3018948\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id2628823\">For the following exercises, solve the rational exponent equation. Use factoring where necessary.<\/p>\n\n<div id=\"fs-id2628826\">\n<div id=\"fs-id2628827\">\n<p id=\"fs-id2628828\">[latex]{x}^{\\frac{2}{3}}=16[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1973815\">\n<div id=\"fs-id1973816\">\n<p id=\"fs-id1973817\">[latex]{x}^{\\frac{3}{4}}=27[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2426775\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2426775\"]\n<p id=\"fs-id2426775\">[latex]x=81[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2432476\">\n<div id=\"fs-id2432477\">\n<p id=\"fs-id2432478\">[latex]2{x}^{\\frac{1}{2}}-{x}^{\\frac{1}{4}}=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id3040226\">\n<div id=\"fs-id3040227\">\n<p id=\"fs-id1296521\">[latex]{\\left(x-1\\right)}^{\\frac{3}{4}}=8[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1934614\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1934614\"]\n<p id=\"fs-id1934614\">[latex]x=17[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id3060328\">\n<div id=\"fs-id3060329\">\n<p id=\"fs-id3060330\">[latex]{\\left(x+1\\right)}^{\\frac{2}{3}}=4[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1763592\">\n<div id=\"fs-id1763593\">\n<p id=\"fs-id1763594\">[latex]{x}^{\\frac{2}{3}}-5{x}^{\\frac{1}{3}}+6=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1239690\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1239690\"]\n<p id=\"fs-id1239690\">[latex]x=8, x=27[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1709842\">\n<div id=\"fs-id1709843\">\n<p id=\"fs-id1709844\">[latex]{x}^{\\frac{7}{3}}-3{x}^{\\frac{4}{3}}-4{x}^{\\frac{1}{3}}=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id2382870\">For the following exercises, solve the following polynomial equations by grouping and factoring.<\/p>\n\n<div id=\"fs-id2382874\">\n<div id=\"fs-id2382875\">\n<p id=\"fs-id2382876\">[latex]{x}^{3}+2{x}^{2}-x-2=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1396320\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1396320\"]\n<p id=\"fs-id1396320\">[latex]x=-2,1,-1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id3111192\">\n<div id=\"fs-id3111193\">\n<p id=\"fs-id3111194\">[latex]3{x}^{3}-6{x}^{2}-27x+54=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1393262\">\n<div id=\"fs-id1393263\">\n<p id=\"fs-id1393264\">[latex]4{y}^{3}-9y=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1567745\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1567745\"]\n<p id=\"fs-id1567745\">[latex]y=0, \\frac{3}{2}, \\frac{-3}{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2529330\">\n<div id=\"fs-id2529331\">\n<p id=\"fs-id2529332\">[latex]{x}^{3}+3{x}^{2}-25x-75=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1582399\">\n<div id=\"fs-id1582400\">\n<p id=\"fs-id1582402\">[latex]{m}^{3}+{m}^{2}-m-1=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2645204\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2645204\"]\n<p id=\"fs-id2645204\">[latex]m=1,-1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id586482\">\n<div id=\"fs-id586483\">\n<p id=\"fs-id2998491\">[latex]2{x}^{5}-14{x}^{3}=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2643100\">\n<div id=\"fs-id2643101\">\n<p id=\"fs-id2643102\">[latex]5{x}^{3}+45x=2{x}^{2}+18[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1337966\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1337966\"]\n<p id=\"fs-id1337966\">[latex]x=\\frac{2}{5},\u00b13i[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id2917413\">For the following exercises, solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions.<\/p>\n\n<div id=\"fs-id2917418\">\n<div id=\"fs-id2294086\">\n<p id=\"fs-id2294087\">[latex]\\sqrt{3x-1}-2=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2332670\">\n<div id=\"fs-id2332671\">\n<p id=\"fs-id2332672\">[latex]\\sqrt{x-7}=5[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2018882\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2018882\"]\n<p id=\"fs-id2018882\">[latex]x=32[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2375964\">\n<div id=\"fs-id2375966\">\n<p id=\"fs-id2375967\">[latex]\\sqrt{x-1}=x-7[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2486495\">\n<div id=\"fs-id2486496\">\n<p id=\"fs-id2486497\">[latex]\\sqrt{3t+5}=7[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2385196\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2385196\"]\n<p id=\"fs-id2385196\">[latex]t=\\frac{44}{3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1551969\">\n<div id=\"fs-id1551970\">\n<p id=\"fs-id1551971\">[latex]\\sqrt{t+1}+9=7[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id3256844\">\n<div id=\"fs-id3256845\">\n<p id=\"fs-id3256846\">[latex]\\sqrt{12-x}=x[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1333069\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1333069\"]\n<p id=\"fs-id1333069\">[latex]x=3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1357710\">\n<div id=\"fs-id1357711\">\n<p id=\"fs-id1357712\">[latex]\\sqrt{2x+3}-\\sqrt{x+2}=2[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1441558\">\n<div id=\"fs-id1441559\">\n<p id=\"fs-id1441560\">[latex]\\sqrt{3x+7}+\\sqrt{x+2}=1[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2383872\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2383872\"]\n<p id=\"fs-id2383872\">[latex]x=-2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1951699\">\n<p id=\"fs-id1951700\">[latex]\\sqrt{2x+3}-\\sqrt{x+1}=1[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1786258\">For the following exercises, solve the equation involving absolute value.<\/p>\n\n<div id=\"fs-id1150948\">\n<div id=\"fs-id1150949\">\n<p id=\"fs-id1150950\">[latex]|3x-4|=8[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2718106\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2718106\"]\n<p id=\"fs-id2718106\">[latex]x=4,\\frac{-4}{3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2307448\">\n<div id=\"fs-id2307449\">\n<p id=\"fs-id2307450\">[latex]|2x-3|=-2[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2410873\">\n<div id=\"fs-id2410874\">\n<p id=\"fs-id2410875\">[latex]|1-4x|-1=5[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2818662\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2818662\"]\n<p id=\"fs-id2818662\">[latex]x=\\frac{-5}{4},\\frac{7}{4}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2428664\">\n<div id=\"fs-id2428665\">\n<p id=\"fs-id2428666\">[latex]|4x+1|-3=6[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1717824\">\n<div id=\"fs-id1717825\">\n<p id=\"fs-id1717826\">[latex]|2x-1|-7=-2[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1780317\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1780317\"]\n<p id=\"fs-id1780317\">[latex]x=3,-2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id3165049\">\n<div id=\"fs-id3165050\">\n<p id=\"fs-id3165051\">[latex]|2x+1|-2=-3[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2382990\">\n<div id=\"fs-id2382991\">\n<p id=\"fs-id2382992\">[latex]|x+5|=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2390987\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2390987\"]\n<p id=\"fs-id2390987\">[latex]x=-5[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2643356\">\n<div id=\"fs-id2643357\">\n<p id=\"fs-id2643358\">[latex]-|2x+1|=-3[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id2665057\">For the following exercises, solve the equation by identifying the quadratic form. Use a substitute variable and find all real solutions by factoring.<\/p>\n\n<div id=\"fs-id2665062\">\n<div id=\"fs-id2665063\">\n<p id=\"fs-id2386875\">[latex]{x}^{4}-10{x}^{2}+9=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id3126793\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id3126793\"]\n<p id=\"fs-id3126793\">[latex]x=1,-1,3,-3[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1715518\">\n<div id=\"fs-id1715519\">\n<p id=\"fs-id1715520\">[latex]4{\\left(t-1\\right)}^{2}-9\\left(t-1\\right)=-2[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1577029\">\n<div id=\"fs-id2029121\">\n<p id=\"fs-id2029122\">[latex]{\\left({x}^{2}-1\\right)}^{2}+\\left({x}^{2}-1\\right)-12=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id3042785\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id3042785\"]\n<p id=\"fs-id3042785\">[latex]x=2,-2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1418519\">\n<div id=\"fs-id1418520\">\n<p id=\"fs-id1418521\">[latex]{\\left(x+1\\right)}^{2}-8\\left(x+1\\right)-9=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2797000\">\n<div id=\"fs-id2797001\">\n<p id=\"fs-id2797002\">[latex]{\\left(x-3\\right)}^{2}-4=0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1518893\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1518893\"]\n<p id=\"fs-id1518893\">[latex]x=1,5[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2390141\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<p id=\"fs-id2390146\">For the following exercises, solve for the unknown variable.<\/p>\n\n<div>\n<div>\n<p id=\"fs-id2390152\">[latex]{x}^{-2}-{x}^{-1}-12=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1726737\">\n<div id=\"fs-id1726738\">\n<p id=\"fs-id1726739\">[latex]\\sqrt{{|x|}^{2}}=x[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2412232\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2412232\"]\n<p id=\"fs-id2412232\">All real numbers<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2412235\">\n<div id=\"fs-id2412236\">\n<p id=\"fs-id2412237\">[latex]{t}^{10}-{t}^{5}+1=0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1276816\">\n<div id=\"fs-id1276817\">\n<p id=\"fs-id1276818\">[latex]|{x}^{2}+2x-36|=12[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2699615\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2699615\"]\n<p id=\"fs-id2699615\">[latex]x=4,6,-6,-8[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1976510\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<p id=\"fs-id1976516\">For the following exercises, use the model for the period of a pendulum,[latex]\\,T,[\/latex]such that[latex]\\,T=2\\pi \\sqrt{\\frac{L}{g}},[\/latex]where the length of the pendulum is <em>L<\/em> and the acceleration due to gravity is[latex]\\,g.[\/latex]<\/p>\n\n<div id=\"fs-id2500088\">\n<div id=\"fs-id2500089\">\n<p id=\"fs-id2500090\">If the acceleration due to gravity is 9.8 m\/s<sup>2<\/sup> and the period equals 1 s, find the length to the nearest cm (100 cm = 1 m).<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2947584\">\n<div id=\"fs-id2947585\">\n<p id=\"fs-id1762772\">If the gravity is 32 ft\/s<sup>2<\/sup> and the period equals 1 s, find the length to the nearest in. (12 in. = 1 ft). Round your answer to the nearest in.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id3207106\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id3207106\"]\n<p id=\"fs-id3207106\">10 in.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1591913\">For the following exercises, use a model for body surface area, BSA, such that[latex]\\,BSA=\\sqrt{\\frac{wh}{3600}},[\/latex]where <em>w<\/em> = weight in kg and <em>h<\/em> = height in cm.<\/p>\n\n<div>\n<div id=\"fs-id3070468\">\n<p id=\"fs-id3070470\">Find the height of a 72-kg female to the nearest cm whose[latex]\\,BSA=1.8.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1591172\">\n<div id=\"fs-id1591174\">\n<p id=\"fs-id1591175\">Find the weight of a 177-cm male to the nearest kg whose[latex]\\,BSA=2.1.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2290740\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2290740\"]\n<p id=\"fs-id2290740\">90 kg<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id2290748\">\n \t<dt>absolute value equation<\/dt>\n \t<dd id=\"fs-id2290751\">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-id1333937\">\n \t<dt>equations in quadratic form<\/dt>\n \t<dd id=\"fs-id1333940\">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-id1333945\">\n \t<dt>extraneous solutions<\/dt>\n \t<dd id=\"fs-id1333948\">any solutions obtained that are not valid in the original equation<\/dd>\n<\/dl>\n<dl id=\"fs-id1547816\">\n \t<dt>polynomial equation<\/dt>\n \t<dd id=\"fs-id1547819\">an equation containing a string of terms including numerical coefficients and variables raised to whole-number exponents<\/dd>\n<\/dl>\n<dl id=\"fs-id1547823\">\n \t<dt>radical equation<\/dt>\n \t<dd id=\"fs-id1547827\">an equation containing at least one radical term where the variable is part of the radicand<\/dd>\n<\/dl>\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section you will:<\/p>\n<ul>\n<li>Solve equations involving rational exponents.<\/li>\n<li>Solve equations using factoring.<\/li>\n<li>Solve radical equations.<\/li>\n<li>Solve absolute value equations.<\/li>\n<li>Solve other types of equations.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1519532\">We have solved linear equations, rational equations, and quadratic equations using several methods. However, there are many other types of equations, and we will investigate a few more types in this section. We will look at equations involving rational exponents, polynomial equations, radical equations, absolute value equations, equations in quadratic form, and some rational equations that can be transformed into quadratics. Solving any equation, however, employs the same basic algebraic rules. We will learn some new techniques as they apply to certain equations, but the algebra never changes.<\/p>\n<div id=\"fs-id2485804\" class=\"bc-section section\">\n<h3>Solving Equations Involving Rational Exponents<\/h3>\n<p id=\"fs-id1274976\">Rational exponents are exponents that are fractions, where the numerator is a power and the denominator is a root. For example,[latex]\\,{16}^{\\frac{1}{2}}\\,[\/latex]is another way of writing[latex]\\,\\sqrt{16};[\/latex][latex]{8}^{\\frac{1}{3}}\\,[\/latex]is another way of writing[latex]\\text{\u200b}\\,\\sqrt[3]{8}.\\,[\/latex]The ability to work with rational exponents is a useful skill, as it is highly applicable in calculus.<\/p>\n<p id=\"fs-id1723049\">We can solve equations in which a variable is raised to a rational exponent by raising both sides of the equation to the reciprocal of the exponent. The reason we raise the equation to the reciprocal of the exponent is because we want to eliminate the exponent on the variable term, and a number multiplied by its reciprocal equals 1. For example,[latex]\\,\\frac{2}{3}\\left(\\frac{3}{2}\\right)=1,[\/latex][latex]3\\left(\\frac{1}{3}\\right)=1,[\/latex]and so on.<\/p>\n<div id=\"fs-id1319605\" class=\"textbox key-takeaways\">\n<h3>Rational Exponents<\/h3>\n<p id=\"fs-id1532274\">A rational exponent indicates a power in the numerator and a root in the denominator. There are multiple ways of writing an expression, a variable, or a number with a rational exponent:<\/p>\n<div id=\"fs-id1512276\" class=\"unnumbered aligncenter\">[latex]{a}^{\\frac{m}{n}}={\\left({a}^{\\frac{1}{n}}\\right)}^{m}={\\left({a}^{m}\\right)}^{\\frac{1}{n}}=\\sqrt[n]{{a}^{m}}={\\left(\\sqrt[n]{a}\\right)}^{m}[\/latex]<\/div>\n<\/div>\n<div id=\"Example_02_06_01\" class=\"textbox examples\">\n<div id=\"fs-id1458449\">\n<div id=\"fs-id2592723\">\n<h3>Evaluating a Number Raised to a Rational Exponent<\/h3>\n<p id=\"fs-id1239738\">Evaluate[latex]\\,{8}^{\\frac{2}{3}}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2422284\">Whether we take the root first or the power first depends on the number. It is easy to find the cube root of 8, so rewrite[latex]\\,{8}^{\\frac{2}{3}}\\,[\/latex]as[latex]\\,{\\left({8}^{\\frac{1}{3}}\\right)}^{2}.[\/latex]<\/p>\n<div id=\"fs-id1227998\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill {\\left({8}^{\\frac{1}{3}}\\right)}^{2}& =\\hfill & {\\left(2\\right)}^{2}\\hfill \\\\ & =& 4\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1514351\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_06_01\">\n<div id=\"fs-id1336704\">\n<p id=\"fs-id2400251\">Evaluate[latex]\\,{64}^{-\\frac{1}{3}}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2434546\">[latex]\\frac{1}{4}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_02_06_02\" class=\"textbox examples\">\n<div id=\"fs-id2385021\">\n<div id=\"fs-id3264211\">\n<h3>Solve the Equation Including a Variable Raised to a Rational Exponent<\/h3>\n<p id=\"fs-id1198214\">Solve the equation in which a variable is raised to a rational exponent:[latex]\\,{x}^{\\frac{5}{4}}=32.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1569151\">The way to remove the exponent on <em>x<\/em> is by raising both sides of the equation to a power that is the reciprocal of[latex]\\,\\frac{5}{4},[\/latex]which is[latex]\\,\\frac{4}{5}.[\/latex]<\/p>\n<div id=\"fs-id1277511\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cccc}\\hfill {x}^{\\frac{5}{4}}& =& 32\\hfill & \\\\ \\hfill {\\left({x}^{\\frac{5}{4}}\\right)}^{\\frac{4}{5}}& =& {\\left(32\\right)}^{\\frac{4}{5}}\\hfill & \\\\ \\hfill x& =& {\\left(2\\right)}^{4}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{The fifth root of 32 is 2.}\\hfill \\\\ & =& 16\\hfill & \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1759744\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_06_02\">\n<div id=\"fs-id2293541\">\n<p id=\"fs-id1314798\">Solve the equation[latex]\\,{x}^{\\frac{3}{2}}=125.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]25[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_02_06_03\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1335480\">\n<h3>Solving an Equation Involving Rational Exponents and Factoring<\/h3>\n<p id=\"fs-id1269017\">Solve[latex]\\,3{x}^{\\frac{3}{4}}={x}^{\\frac{1}{2}}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>This equation involves rational exponents as well as factoring rational exponents. Let us take this one step at a time. First, put the variable terms on one side of the equal sign and set the equation equal to zero.<\/p>\n<div id=\"fs-id1400272\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill 3{x}^{\\frac{3}{4}}-\\left({x}^{\\frac{1}{2}}\\right)& =& {x}^{\\frac{1}{2}}-\\left({x}^{\\frac{1}{2}}\\right)\\hfill \\\\ \\hfill 3{x}^{\\frac{3}{4}}-{x}^{\\frac{1}{2}}& =& 0\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1368636\">Now, it looks like we should factor the left side, but what do we factor out? We can always factor the term with the lowest exponent. Rewrite[latex]\\,{x}^{\\frac{1}{2}}\\,[\/latex]as[latex]\\,{x}^{\\frac{2}{4}}.\\,[\/latex]Then, factor out[latex]\\,{x}^{\\frac{2}{4}}\\,[\/latex]from both terms on the left.<\/p>\n<div id=\"fs-id1441452\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill 3{x}^{\\frac{3}{4}}-{x}^{\\frac{2}{4}}& =& 0\\hfill \\\\ \\hfill {x}^{\\frac{2}{4}}\\left(3{x}^{\\frac{1}{4}}-1\\right)& =& 0\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1933337\">Where did[latex]\\,{x}^{\\frac{1}{4}}\\,[\/latex]come from? Remember, when we multiply two numbers with the same base, we add the exponents. Therefore, if we multiply[latex]\\,{x}^{\\frac{2}{4}}\\,[\/latex]back in using the distributive property, we get the expression we had before the factoring, which is what should happen. We need an exponent such that when added to[latex]\\,\\frac{2}{4}\\,[\/latex]equals[latex]\\,\\frac{3}{4}.\\,[\/latex]Thus, the exponent on <em>x <\/em>in the parentheses is[latex]\\,\\frac{1}{4}.\\,[\/latex]<\/p>\n<p id=\"fs-id1891109\">Let us continue. Now we have two factors and can use the zero factor theorem.<\/p>\n<div id=\"fs-id1422353\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cccc}\\hfill {x}^{\\frac{2}{4}}\\left(3{x}^{\\frac{1}{4}}-1\\right)& =& 0\\hfill & \\\\ \\hfill {x}^{\\frac{2}{4}}& =& 0\\hfill & \\\\ \\hfill x& =& 0\\hfill & \\\\ \\hfill 3{x}^{\\frac{1}{4}}-1& =& 0\\hfill & \\\\ \\hfill 3{x}^{\\frac{1}{4}}& =& 1\\hfill & \\\\ \\hfill {x}^{\\frac{1}{4}}& =& \\frac{1}{3}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Divide both sides by 3}.\\hfill \\\\ \\hfill {\\left({x}^{\\frac{1}{4}}\\right)}^{4}& =& {\\left(\\frac{1}{3}\\right)}^{4}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Raise both sides to the reciprocal of }\\frac{1}{4}.\\hfill \\\\ \\hfill x& =& \\frac{1}{81}\\hfill & \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2957928\">The two solutions are [latex]\\,0[\/latex] and [latex]\\frac{1}{81}.[\/latex]<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1267637\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_06_03\">\n<div id=\"fs-id1411367\">\n<p id=\"fs-id1441991\">Solve:[latex]\\,{\\left(x+5\\right)}^{\\frac{3}{2}}=8.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1846504\">[latex]\\left\\{-1\\right\\}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2945385\" class=\"bc-section section\">\n<h3>Solving Equations Using Factoring<\/h3>\n<p id=\"fs-id2521696\">We have used factoring to solve quadratic equations, but it is a technique that we can use with many types of polynomial equations, which are equations that contain a string of terms including numerical coefficients and variables. When we are faced with an equation containing polynomials of degree higher than 2, we can often solve them by factoring.<\/p>\n<div id=\"fs-id1786047\" class=\"textbox key-takeaways\">\n<h3>Polynomial Equations<\/h3>\n<p id=\"fs-id3231810\">A polynomial of degree <em>n <\/em>is an expression of the type<\/p>\n<div id=\"fs-id2905918\" class=\"unnumbered aligncenter\">[latex]{a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+\\,\\,\\,\\cdot \\,\\,\\cdot \\,\\,\\cdot \\,\\,+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[\/latex]<\/div>\n<p id=\"fs-id2444064\">where <em>n<\/em> is a positive integer and[latex]\\,{a}_{n},\\dots ,{a}_{0}\\,[\/latex]are real numbers and[latex]\\,{a}_{n}\\ne 0.[\/latex]<\/p>\n<p id=\"fs-id2931289\">Setting the polynomial equal to zero gives a polynomial equation. The total number of solutions (real and complex) to a polynomial equation is equal to the highest exponent <em>n<\/em>.<\/p>\n<\/div>\n<div id=\"Example_02_06_04\" class=\"textbox examples\">\n<div id=\"fs-id1996696\">\n<div id=\"fs-id2501149\">\n<h3>Solving a Polynomial by Factoring<\/h3>\n<p id=\"fs-id1391817\">Solve the polynomial by factoring:[latex]\\,5{x}^{4}=80{x}^{2}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>First, set the equation equal to zero. Then factor out what is common to both terms, the GCF.<\/p>\n<div id=\"fs-id2502503\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill 5{x}^{4}-80{x}^{2}& =& 0\\hfill \\\\ \\hfill 5{x}^{2}\\left({x}^{2}-16\\right)& =& 0\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1924982\">Notice that we have the difference of squares in the factor[latex]\\,{x}^{2}-16,[\/latex]which we will continue to factor and obtain two solutions. The first term,[latex]\\,5{x}^{2},[\/latex]generates, technically, two solutions as the exponent is 2, but they are the same solution.<\/p>\n<div id=\"fs-id2824827\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill 5{x}^{2}& =& 0\\hfill \\\\ \\hfill x& =& 0\\hfill \\\\ \\hfill {x}^{2}-16& =& 0\\hfill \\\\ \\hfill \\left(x-4\\right)\\left(x+4\\right)& =& 0\\hfill \\\\ \\hfill x& =& 4\\hfill \\\\ \\hfill x& =& -4\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2638784\">The solutions are [latex]\\,0\\text{ (double solution),}[\/latex][latex]4,[\/latex] and [latex]\\,-4.[\/latex]<\/details>\n<\/p>\n<\/div>\n<div id=\"fs-id1693367\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id768264\">We can see the solutions on the graph in <a class=\"autogenerated-content\" href=\"#Figure_02_06_001\">(Figure)<\/a>. The <em>x-<\/em>coordinates of the points where the graph crosses the <em>x-<\/em>axis are the solutions\u2014the <em>x-<\/em>intercepts. Notice on the graph that at the solution[latex]\\,0,[\/latex]the graph touches the <em>x-<\/em>axis and bounces back. It does not cross the <em>x-<\/em>axis. This is typical of double solutions.<\/p>\n<div id=\"Figure_02_06_001\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133613\/CNX_CAT_Figure_02_06_001.jpg\" alt=\"Coordinate plane with the x-axis ranging from negative 5 to 5 and the y-axis ranging from negative 400 to 500 in intervals of 100. The function five times x to the fourth power minus eighty x squared equals zero is graphed along with the points (negative 4,0), (0,0), and (4,0).\" width=\"487\" height=\"401\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_06_04\">\n<div id=\"fs-id2496351\">\n<p id=\"fs-id3040346\">Solve by factoring:[latex]\\,12{x}^{4}=3{x}^{2}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1244467\">[latex]x=0,[\/latex][latex]x=\\frac{1}{2},[\/latex][latex]x=-\\frac{1}{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_02_06_05\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1007381\">\n<h3>Solve a Polynomial by Grouping<\/h3>\n<p id=\"fs-id2501656\">Solve a polynomial by grouping:[latex]\\,{x}^{3}+{x}^{2}-9x-9=0.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1422651\">This polynomial consists of 4 terms, which we can solve by grouping. Grouping procedures require factoring the first two terms and then factoring the last two terms. If the factors in the parentheses are identical, we can continue the process and solve, unless more factoring is suggested.<\/p>\n<div id=\"fs-id2516826\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill {x}^{3}+{x}^{2}-9x-9& =& 0\\hfill \\\\ \\hfill {x}^{2}\\left(x+1\\right)-9\\left(x+1\\right)& =& 0\\hfill \\\\ \\hfill \\left({x}^{2}-9\\right)\\left(x+1\\right)& =& 0\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1408391\">The grouping process ends here, as we can factor[latex]\\,{x}^{2}-9\\,[\/latex]<br \/>\nusing the difference of squares formula.<\/p>\n<div id=\"fs-id1402146\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\left({x}^{2}-9\\right)\\left(x+1\\right)& =& 0\\hfill \\\\ \\hfill \\left(x-3\\right)\\left(x+3\\right)\\left(x+1\\right)& =& 0\\hfill \\\\ \\hfill x& =& 3\\hfill \\\\ \\hfill x& =& -3\\hfill \\\\ \\hfill x& =& -1\\hfill \\end{array}[\/latex]<\/div>\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19133616\/CNX_CAT_Figure_02_06_002.jpg\" alt=\"Coordinate plane with the x-axis ranging from negative 5 to 5 and the y-axis ranging from negative 30 to 20 in intervals of 5. The function x cubed plus x squared minus nine times x minus nine equals zero is graphed along with the points (negative 3,0), (negative 1,0), and (3,0).\" width=\"487\" height=\"438\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id2528732\">The solutions are [latex]3,[\/latex][latex]-3,[\/latex] and [latex]\\,-1.\\,[\/latex]Note that the highest exponent is 3 and we obtained 3 solutions. We can see the solutions, the <em>x-<\/em>intercepts, on the graph in <a class=\"autogenerated-content\" href=\"#Figure_02_06_002\">(Figure)<\/a>.<span id=\"fs-id2304259\"><\/span><\/details>\n<\/p>\n<\/div>\n<div id=\"fs-id2504109\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1243628\">We looked at solving quadratic equations by factoring when the leading coefficient is 1. When the leading coefficient is not 1, we solved by grouping. Grouping requires four terms, which we obtained by splitting the linear term of quadratic equations. We can also use grouping for some polynomials of degree higher than 2, as we saw here, since there were already four terms.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2506398\" class=\"bc-section section\">\n<h3>Solving Radical Equations<\/h3>\n<p id=\"fs-id1846458\"><strong>Radical equations<\/strong> are equations that contain variables in the <span class=\"no-emphasis\">radicand<\/span> (the expression under a radical symbol), such as<\/p>\n<div id=\"fs-id2781660\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\sqrt{3x+18}& =& x\\hfill \\\\ \\hfill \\sqrt{x+3}& =& x-3\\hfill \\\\ \\hfill \\sqrt{x+5}-\\sqrt{x-3}& =& 2\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1719179\">Radical equations may have one or more radical terms, and are solved by eliminating each radical, one at a time. We have to be careful when solving radical equations, as it is not unusual to find extraneous solutions, roots that are not, in fact, solutions to the equation. These solutions are not due to a mistake in the solving method, but result from the process of raising both sides of an equation to a power. However, checking each answer in the original equation will confirm the true solutions.<\/p>\n<div id=\"fs-id2506978\" class=\"textbox key-takeaways\">\n<h3>Radical Equations<\/h3>\n<p id=\"eip-id3052762\">An equation containing terms with a variable in the radicand is called a radical equation.<\/p>\n<\/div>\n<div id=\"fs-id2307719\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1517012\"><strong>Given a radical equation, solve it.<\/strong><\/p>\n<ol id=\"fs-id3143660\" type=\"1\">\n<li>Isolate the radical expression on one side of the equal sign. Put all remaining terms on the other side.<\/li>\n<li>If the radical is a square root, then square both sides of the equation. If it is a cube root, then raise both sides of the equation to the third power. In other words, for an <em>n<\/em>th root radical, raise both sides to the <em>n<\/em>th power. Doing so eliminates the radical symbol.<\/li>\n<li>Solve the remaining equation.<\/li>\n<li>If a radical term still remains, repeat steps 1\u20132.<\/li>\n<li>Confirm solutions by substituting them into the original equation.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_02_06_06\" class=\"textbox examples\">\n<div id=\"fs-id1222816\">\n<div id=\"fs-id1551039\">\n<h3>Solving an Equation with One Radical<\/h3>\n<p id=\"fs-id1520471\">Solve[latex]\\,\\sqrt{15-2x}=x.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1332305\">The radical is already isolated on the left side of the equal side, so proceed to square both sides.<\/p>\n<div id=\"fs-id1919300\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\sqrt{15-2x}& =& x\\hfill \\\\ \\hfill {\\left(\\sqrt{15-2x}\\right)}^{2}& =& {\\left(x\\right)}^{2}\\hfill \\\\ \\hfill 15-2x& =& {x}^{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1895386\">We see that the remaining equation is a quadratic. Set it equal to zero and solve.<\/p>\n<div id=\"fs-id1905278\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill 0& =& {x}^{2}+2x-15\\hfill \\\\ & =& \\left(x+5\\right)\\left(x-3\\right)\\hfill \\\\ \\hfill x& =& -5\\hfill \\\\ \\hfill x& =& 3\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1257720\">The proposed solutions are [latex]-5\\,[\/latex] and [latex]3.\\,[\/latex]Let us check each solution back in the original equation. First, check[latex]\\,x=-5.[\/latex]<\/p>\n<div id=\"fs-id1463741\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\sqrt{15-2x}& =& x\\hfill \\\\ \\hfill \\sqrt{15-2\\left(-5\\right)}& =& -5\\hfill \\\\ \\hfill \\sqrt{25}& =& -5\\hfill \\\\ \\hfill 5& \\ne & -5\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2628354\">This is an extraneous solution. While no mistake was made solving the equation, we found a solution that does not satisfy the original equation.<\/p>\n<p id=\"fs-id1417225\">Check[latex]\\,x=3.[\/latex]<\/p>\n<div id=\"fs-id1515949\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\sqrt{15-2x}& =& x\\hfill \\\\ \\hfill \\sqrt{15-2\\left(3\\right)}& =& 3\\hfill \\\\ \\hfill \\sqrt{9}& =& 3\\hfill \\\\ \\hfill 3& =& 3\\hfill \\end{array}[\/latex]<\/div>\n<p>The solution is [latex]\\,3.[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1441693\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_06_05\">\n<div id=\"fs-id1583879\">\n<p id=\"fs-id1583880\">Solve the radical equation:[latex]\\,\\sqrt{x+3}=3x-1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1535940\">[latex]x=1;[\/latex]extraneous solution[latex]\\,x=-\\frac{2}{9}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_02_06_07\" class=\"textbox examples\">\n<div id=\"fs-id1720874\">\n<div id=\"fs-id2499586\">\n<h3>Solving a Radical Equation Containing Two Radicals<\/h3>\n<p id=\"fs-id1410869\">Solve[latex]\\,\\sqrt{2x+3}+\\sqrt{x-2}=4.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1929508\">As this equation contains two radicals, we isolate one radical, eliminate it, and then isolate the second radical.<\/p>\n<div id=\"fs-id2268062\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cccc}\\hfill \\sqrt{2x+3}+\\sqrt{x-2}& =& 4\\hfill & \\\\ \\hfill \\sqrt{2x+3}& =& 4-\\sqrt{x-2}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Subtract }\\sqrt{x-2}\\text{ from both sides}.\\hfill \\\\ \\hfill {\\left(\\sqrt{2x+3}\\right)}^{2}& =& {\\left(4-\\sqrt{x-2}\\right)}^{2}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Square both sides}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1548786\">Use the perfect square formula to expand the right side:[latex]\\,{\\left(a-b\\right)}^{2}={a}^{2}-2ab+{b}^{2}.[\/latex]<\/p>\n<div id=\"fs-id2506054\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cccc}\\hfill 2x+3& =& {\\left(4\\right)}^{2}-2\\left(4\\right)\\sqrt{x-2}+{\\left(\\sqrt{x-2}\\right)}^{2}\\hfill & \\\\ \\hfill 2x+3& =& 16-8\\sqrt{x-2}+\\left(x-2\\right)\\hfill & \\\\ \\hfill 2x+3& =& 14+x-8\\sqrt{x-2}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Combine like terms}.\\hfill \\\\ \\hfill x-11& =& -8\\sqrt{x-2}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Isolate the second radical}.\\hfill \\\\ \\hfill {\\left(x-11\\right)}^{2}& =& {\\left(-8\\sqrt{x-2}\\right)}^{2}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Square both sides}.\\hfill \\\\ \\hfill {x}^{2}-22x+121& =& 64\\left(x-2\\right)\\hfill & \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1175462\">Now that both radicals have been eliminated, set the quadratic equal to zero and solve.<\/p>\n<div id=\"fs-id2946619\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cccc}\\hfill {x}^{2}-22x+121& =& 64x-128\\hfill & \\\\ \\hfill {x}^{2}-86x+249& =& 0\\hfill & \\\\ \\hfill \\left(x-3\\right)\\left(x-83\\right)& =& 0\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Factor and solve}.\\hfill \\\\ \\hfill x& =& 3\\hfill & \\\\ \\hfill x& =& 83\\hfill & \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1563543\">The proposed solutions are [latex]\\,3\\,[\/latex] and [latex]\\,83.\\,[\/latex]Check each solution in the original equation.<\/p>\n<div id=\"fs-id2655511\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\sqrt{2x+3}+\\sqrt{x-2}& =& 4\\hfill \\\\ \\hfill \\sqrt{2x+3}& =& 4-\\sqrt{x-2}\\hfill \\\\ \\hfill \\sqrt{2\\left(3\\right)+3}& =& 4-\\sqrt{\\left(3\\right)-2}\\hfill \\\\ \\hfill \\sqrt{9}& =& 4-\\sqrt{1}\\hfill \\\\ \\hfill 3& =& 3\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1840544\">One solution is [latex]\\,3.[\/latex]<\/p>\n<p id=\"fs-id1918601\">Check[latex]\\,x=83.[\/latex]<\/p>\n<div id=\"fs-id1941177\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\sqrt{2x+3}+\\sqrt{x-2}& =& 4\\hfill \\\\ \\hfill \\sqrt{2x+3}& =& 4-\\sqrt{x-2}\\hfill \\\\ \\hfill \\sqrt{2\\left(83\\right)+3}& =& 4-\\sqrt{\\left(83-2\\right)}\\hfill \\\\ \\hfill \\sqrt{169}& =& 4-\\sqrt{81}\\hfill \\\\ \\hfill 13& \\ne & -5\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2293780\">The only solution is [latex]\\,3.\\,[\/latex]We see that[latex]\\,x=83\\,[\/latex]is an extraneous solution.<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2430499\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_06_06\">\n<div id=\"fs-id2702970\">\n<p id=\"fs-id2429740\">Solve the equation with two radicals:[latex]\\,\\sqrt{3x+7}+\\sqrt{x+2}=1.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1951350\">[latex]x=-2;[\/latex]extraneous solution[latex]\\,x=-1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id3240737\" class=\"bc-section section\">\n<h3>Solving an Absolute Value Equation<\/h3>\n<p id=\"fs-id1335626\">Next, we will learn how to solve an <span class=\"no-emphasis\">absolute value equation<\/span>. To solve an equation such as[latex]\\,|2x-6|=8,[\/latex]we notice that the absolute value will be equal to 8 if the quantity inside the absolute value bars is[latex]\\,8\\,[\/latex]or[latex]\\,-8.\\,[\/latex]This leads to two different equations we can solve independently.<\/p>\n<div id=\"fs-id2638694\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccccccc}\\hfill 2x-6& =& 8\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{or}\\phantom{\\rule{2em}{0ex}}& \\hfill 2x-6& =& -8\\hfill \\\\ \\hfill 2x& =& 14& & \\hfill 2x& =& -2\\hfill \\\\ \\hfill x& =& 7\\hfill & & \\hfill x& =& -1\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1280820\">Knowing how to solve problems involving absolute value functions is useful. For example, we may need to identify numbers or points on a line that are at a specified distance from a given reference point.<\/p>\n<div id=\"fs-id2918915\" class=\"textbox key-takeaways\">\n<h3>Absolute Value Equations<\/h3>\n<p id=\"fs-id1542481\">The absolute value of <em>x <\/em>is written as[latex]\\,|x|.\\,[\/latex]It has the following properties:<\/p>\n<div id=\"fs-id2434805\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{If }x\\ge 0,\\text{ then }|x|=x.\\hfill \\\\ \\text{If }x<0,\\text{ then }|x|=-x.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2905392\">For real numbers[latex]\\,A\\,[\/latex]and[latex]\\,B,[\/latex]an equation of the form[latex]\\,|A|=B,[\/latex]with[latex]\\,B\\ge 0,[\/latex]will have solutions when[latex]\\,A=B\\,[\/latex]or[latex]\\,A=-B.\\,[\/latex]If[latex]\\,B<0,[\/latex]the equation[latex]\\,|A|=B\\,[\/latex]has no solution.<\/p>\n<p id=\"fs-id1408140\">An absolute value equation in the form[latex]\\,|ax+b|=c\\,[\/latex]has the following properties:<\/p>\n<div id=\"fs-id2655133\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{If }c<0,|ax+b|=c\\text{ has no solution}.\\hfill \\\\ \\text{If }c=0,|ax+b|=c\\text{ has one solution}.\\hfill \\\\ \\text{If }c>0,|ax+b|=c\\text{ has two solutions}.\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id3038796\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1793504\"><strong>Given an absolute value equation, solve it.<\/strong><\/p>\n<ol id=\"fs-id2938958\" type=\"1\">\n<li>Isolate the absolute value expression on one side of the equal sign.<\/li>\n<li>If[latex]\\,c>0,[\/latex]write and solve two equations:[latex]\\,ax+b=c\\,[\/latex]and[latex]\\,ax+b=-c.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_02_06_08\" class=\"textbox examples\">\n<div id=\"fs-id3105697\">\n<div id=\"fs-id1836188\">\n<h3>Solving Absolute Value Equations<\/h3>\n<p id=\"fs-id1836194\">Solve the following absolute value equations:<\/p>\n<ul id=\"fs-id2370595\">\n<li>(a) [latex]|6x+4|=8[\/latex]<\/li>\n<li>(b) [latex]|3x+4|=-9[\/latex]<\/li>\n<li>(c) [latex]|3x-5|-4=6[\/latex]<\/li>\n<li>(d) [latex]|-5x+10|=0[\/latex]<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<ul id=\"fs-id2683629\">\n<li>\n<p id=\"fs-id1832720\">(a) [latex]|6x+4|=8[\/latex]<\/p>\n<p id=\"fs-id2389886\">Write two equations and solve each:<\/p>\n<div id=\"eip-id1170402604810\" class=\"unnumbered\">[latex]\\begin{array}{ccccccc}\\hfill 6x+4& =& 8\\hfill & \\phantom{\\rule{5em}{0ex}}& \\hfill 6x+4& =& -8\\hfill \\\\ \\hfill 6x& =& 4\\hfill & \\phantom{\\rule{5em}{0ex}}& \\hfill 6x& =& -12\\hfill \\\\ \\hfill x& =& \\frac{2}{3}\\hfill & \\phantom{\\rule{5em}{0ex}}& \\hfill x& =& -2\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id3203361\">The two solutions are [latex]\\,\\frac{2}{3}[\/latex] and [latex]-2.[\/latex]<\/p>\n<\/li>\n<li>\n<p id=\"fs-id1879343\">(b) [latex]|3x+4|=-9[\/latex]<\/p>\n<p id=\"fs-id2293707\">There is no solution as an absolute value cannot be negative.<\/p>\n<\/li>\n<li>\n<p id=\"fs-id2888792\">(c) [latex]|3x-5|-4=6[\/latex]<\/p>\n<p id=\"fs-id2436076\">Isolate the absolute value expression and then write two equations.<\/p>\n<div id=\"fs-id3274908\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccccccccc}& & & \\hfill |3x-5|-4& =& 6\\hfill & & & \\\\ & & & \\hfill |3x-5|& =& 10\\hfill & & & \\\\ \\hfill 3x-5& =& 10\\hfill & & & & \\hfill 3x-5& =& -10\\hfill \\\\ \\hfill 3x& =& 15\\hfill & & & & \\hfill 3x& =& -5\\hfill \\\\ \\hfill x& =& 5\\hfill & & & & \\hfill x& =& -\\frac{5}{3}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2498828\">There are two solutions: [latex]\\,5,[\/latex] and [latex]-\\frac{5}{3}.[\/latex]<\/p>\n<\/li>\n<li>\n<p id=\"fs-id1340685\">(d) [latex]|-5x+10|=0[\/latex]<\/p>\n<p>The equation is set equal to zero, so we have to write only one equation.<\/p>\n<div id=\"fs-id2431537\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill -5x+10& =& 0\\hfill \\\\ \\hfill -5x& =& -10\\hfill \\\\ \\hfill x& =& 2\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1351555\">There is one solution: [latex]\\,2.[\/latex]<\/details>\n<\/p>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1535632\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_06_07\">\n<div id=\"fs-id2515512\">\n<p id=\"fs-id2515513\">Solve the absolute value equation:[latex]|1-4x|+8=13.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2385860\">[latex]x=-1,[\/latex][latex]x=\\frac{3}{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2653809\" class=\"bc-section section\">\n<h3>Solving Other Types of Equations<\/h3>\n<p id=\"fs-id3169548\">There are many other types of equations in addition to the ones we have discussed so far. We will see more of them throughout the text. Here, we will discuss equations that are in quadratic form, and rational equations that result in a quadratic.<\/p>\n<div id=\"fs-id1923869\" class=\"bc-section section\">\n<h4>Solving Equations in Quadratic Form<\/h4>\n<p id=\"fs-id1923873\"><strong>Equations in quadratic form <\/strong>are equations with three terms. The first term has a power other than 2. The middle term has an exponent that is one-half the exponent of the leading term. The third term is a constant. We can solve equations in this form as if they were quadratic. A few examples of these equations include[latex]\\,{x}^{4}-5{x}^{2}+4=0,{x}^{6}+7{x}^{3}-8=0,[\/latex]and[latex]\\,{x}^{\\frac{2}{3}}+4{x}^{\\frac{1}{3}}+2=0.\\,[\/latex]In each one, doubling the exponent of the middle term equals the exponent on the leading term. We can solve these equations by substituting a variable for the middle term.<\/p>\n<div id=\"fs-id1834422\" class=\"textbox key-takeaways\">\n<h3>Quadratic Form<\/h3>\n<p id=\"fs-id3094567\">If the exponent on the middle term is one-half of the exponent on the leading term, we have an equation in quadratic form, which we can solve as if it were a quadratic. We substitute a variable for the middle term to solve equations in quadratic form.<\/p>\n<\/div>\n<div id=\"fs-id1954745\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1385726\"><strong>Given an equation quadratic in form, solve it.<\/strong><\/p>\n<ol id=\"fs-id1385730\" type=\"1\">\n<li>Identify the exponent on the leading term and determine whether it is double the exponent on the middle term.<\/li>\n<li>If it is, substitute a variable, such as <em>u<\/em>, for the variable portion of the middle term.<\/li>\n<li>Rewrite the equation so that it takes on the standard form of a quadratic.<\/li>\n<li>Solve using one of the usual methods for solving a quadratic.<\/li>\n<li>Replace the substitution variable with the original term.<\/li>\n<li>Solve the remaining equation.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_02_06_09\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1357816\">\n<h3>Solving a Fourth-degree Equation in Quadratic Form<\/h3>\n<p id=\"fs-id2486573\">Solve this fourth-degree equation:[latex]\\,3{x}^{4}-2{x}^{2}-1=0.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1229229\">This equation fits the main criteria, that the power on the leading term is double the power on the middle term. Next, we will make a substitution for the variable term in the middle. Let[latex]\\,u={x}^{2}.[\/latex]Rewrite the equation in <em>u<\/em>.<\/p>\n<div class=\"unnumbered\">[latex]3{u}^{2}-2u-1=0[\/latex]<\/div>\n<p id=\"fs-id2371956\">Now solve the quadratic.<\/p>\n<div id=\"fs-id2371960\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill 3{u}^{2}-2u-1& =& 0\\hfill \\\\ \\hfill \\left(3u+1\\right)\\left(u-1\\right)& =& 0\\hfill \\end{array}[\/latex]<\/div>\n<p>Solve each factor and replace the original term for <em>u.<\/em><\/p>\n<div id=\"fs-id2797268\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill 3u+1& =& 0\\hfill \\\\ \\hfill 3u& =& -1\\hfill \\\\ \\hfill u& =& -\\frac{1}{3}\\hfill \\\\ \\hfill {x}^{2}& =& -\\frac{1}{3}\\hfill \\\\ \\hfill x& =& \u00b1i\\sqrt{\\frac{1}{3}}\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"fs-id1335933\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill u-1& =& 0\\hfill \\\\ \\hfill u& =& 1\\hfill \\\\ \\hfill {x}^{2}& =& 1\\hfill \\\\ \\hfill x& =& \u00b11\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2980376\">The solutions are [latex]\\,\u00b1i\\sqrt{\\frac{1}{3}}\\,[\/latex] and [latex]\\,\u00b11.[\/latex]<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1929412\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_06_08\">\n<div id=\"fs-id2514218\">\n<p id=\"fs-id2514219\">Solve using substitution:[latex]\\,{x}^{4}-8{x}^{2}-9=0.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1929399\">[latex]x=-3,3,-i,i[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_02_06_10\" class=\"textbox examples\">\n<div id=\"fs-id2444359\">\n<div id=\"fs-id2444361\">\n<h3>Solving an Equation in Quadratic Form Containing a Binomial<\/h3>\n<p id=\"fs-id2029548\">Solve the equation in quadratic form:[latex]\\,{\\left(x+2\\right)}^{2}+11\\left(x+2\\right)-12=0.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2382012\">This equation contains a binomial in place of the single variable. The tendency is to expand what is presented. However, recognizing that it fits the criteria for being in quadratic form makes all the difference in the solving process. First, make a substitution, letting[latex]\\,u=x+2.\\,[\/latex]Then rewrite the equation in <em>u.<\/em><\/p>\n<div id=\"fs-id1929605\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill {u}^{2}+11u-12& =& 0\\hfill \\\\ \\hfill \\left(u+12\\right)\\left(u-1\\right)& =& 0\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1691070\">Solve using the zero-factor property and then replace <em>u<\/em> with the original expression.<\/p>\n<div id=\"fs-id1516932\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill u+12& =& 0\\hfill \\\\ \\hfill u& =& -12\\hfill \\\\ \\hfill x+2& =& -12\\hfill \\\\ \\hfill x& =& -14\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1720759\">The second factor results in<\/p>\n<div id=\"fs-id2485990\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill u-1& =& 0\\hfill \\\\ \\hfill u& =& 1\\hfill \\\\ \\hfill x+2& =& 1\\hfill \\\\ \\hfill x& =& -1\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1943470\">We have two solutions: [latex]\\,-14,[\/latex] and [latex]-1.[\/latex]<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_06_09\">\n<div id=\"fs-id1952007\">\n<p id=\"fs-id1952008\">Solve:[latex]\\,{\\left(x-5\\right)}^{2}-4\\left(x-5\\right)-21=0.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1789429\">[latex]x=2,x=12[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2893053\" class=\"bc-section section\">\n<h4>Solving Rational Equations Resulting in a Quadratic<\/h4>\n<p id=\"fs-id1679917\">Earlier, we solved rational equations. Sometimes, solving a rational equation results in a quadratic. When this happens, we continue the solution by simplifying the quadratic equation by one of the methods we have seen. It may turn out that there is no solution.<\/p>\n<div id=\"Example_02_06_11\" class=\"textbox examples\">\n<div id=\"fs-id1970290\">\n<div id=\"fs-id2508561\">\n<h3>Solving a Rational Equation Leading to a Quadratic<\/h3>\n<p id=\"fs-id2508566\">Solve the following rational equation:[latex]\\,\\frac{-4x}{x-1}+\\frac{4}{x+1}=\\frac{-8}{{x}^{2}-1}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2797287\">We want all denominators in factored form to find the LCD. Two of the denominators cannot be factored further. However,[latex]\\,{x}^{2}-1=\\left(x+1\\right)\\left(x-1\\right).\\,[\/latex]Then, the LCD is[latex]\\,\\left(x+1\\right)\\left(x-1\\right).\\,[\/latex]Next, we multiply the whole equation by the LCD.<\/p>\n<div id=\"fs-id2441856\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\left(x+1\\right)\\left(x-1\\right)\\left[\\frac{-4x}{x-1}+\\frac{4}{x+1}\\right]& =& \\left[\\frac{-8}{\\left(x+1\\right)\\left(x-1\\right)}\\right]\\left(x+1\\right)\\left(x-1\\right)\\hfill \\\\ \\hfill -4x\\left(x+1\\right)+4\\left(x-1\\right)& =& -8\\hfill \\\\ \\hfill -4{x}^{2}-4x+4x-4& =& -8\\hfill \\\\ \\hfill -4{x}^{2}+4& =& 0\\hfill \\\\ \\hfill -4\\left({x}^{2}-1\\right)& =& 0\\hfill \\\\ \\hfill -4\\left(x+1\\right)\\left(x-1\\right)& =& 0\\hfill \\\\ \\hfill x& =& -1\\hfill \\\\ \\hfill x& =& 1\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2980350\">In this case, either solution produces a zero in the denominator in the original equation. Thus, there is no solution.<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2906702\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_02_06_10\">\n<div id=\"fs-id1751154\">\n<p id=\"fs-id1751155\">Solve[latex]\\,\\frac{3x+2}{x-2}+\\frac{1}{x}=\\frac{-2}{{x}^{2}-2x}.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1831304\">[latex]x=-1,[\/latex][latex]x=0[\/latex]is not a solution.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2665007\" class=\"precalculus media\">\n<p id=\"fs-id1822998\">Access these online resources for additional instruction and practice with different types of equations.<\/p>\n<ul id=\"fs-id1823002\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/rateqnosoln\">Rational Equation with no Solution<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/ratexprecpexp\">Solving equations with rational exponents using reciprocal powers<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/radeqsolvepart1\">Solving radical equations part 1 of 2<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/radeqsolvepart2\">Solving radical equations part 2 of 2<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2755534\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id3148376\">\n<li>Rational exponents can be rewritten several ways depending on what is most convenient for the problem. To solve, both sides of the equation are raised to a power that will render the exponent on the variable equal to 1. See <a class=\"autogenerated-content\" href=\"#Example_02_06_01\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_02_06_02\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_02_06_03\">(Figure)<\/a>.<\/li>\n<li>Factoring extends to higher-order polynomials when it involves factoring out the GCF or factoring by grouping. See <a class=\"autogenerated-content\" href=\"#Example_02_06_04\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_02_06_05\">(Figure)<\/a>.<\/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. See <a class=\"autogenerated-content\" href=\"#Example_02_06_06\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_02_06_07\">(Figure)<\/a>.<\/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. See <a class=\"autogenerated-content\" href=\"#Example_02_06_08\">(Figure)<\/a>.<\/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. See <a class=\"autogenerated-content\" href=\"#Example_02_06_09\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_02_06_10\">(Figure)<\/a>.<\/li>\n<li>Solving a rational equation may also lead to a quadratic equation or an equation in quadratic form. See <a class=\"autogenerated-content\" href=\"#Example_02_06_11\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id3146179\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id3146185\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id2028956\">\n<div id=\"fs-id2028957\">\n<p id=\"fs-id2028958\">In a radical equation, what does it mean if a number is an extraneous solution?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2028962\">This is not a solution to the radical equation, it is a value obtained from squaring both sides and thus changing the signs of an equation which has caused it not to be a solution in the original equation.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id3256935\">\n<div id=\"fs-id3256936\">\n<p id=\"fs-id3256937\">Explain why possible solutions <em>must<\/em> be checked in radical equations.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2501234\">\n<div id=\"fs-id2501235\">\n<p id=\"fs-id2501236\">Your friend tries to calculate the value[latex]\\,-{9}^{\\frac{3}{2}}[\/latex]and keeps getting an ERROR message. What mistake is he or she probably making?<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1690245\">He or she is probably trying to enter negative 9, but taking the square root of[latex]\\,-9\\,[\/latex]is not a real number. The negative sign is in front of this, so your friend should be taking the square root of 9, cubing it, and then putting the negative sign in front, resulting in[latex]\\,-27.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1941430\">\n<div id=\"fs-id1941431\">\n<p id=\"fs-id1941432\">Explain why[latex]\\,|2x+5|=-7\\,[\/latex]has no solutions.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1354283\">\n<div id=\"fs-id1354284\">\n<p id=\"fs-id1354285\">Explain how to change a rational exponent into the correct radical expression.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id3018943\">A rational exponent is a fraction: the denominator of the fraction is the root or index number and the numerator is the power to which it is raised.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id3018948\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id2628823\">For the following exercises, solve the rational exponent equation. Use factoring where necessary.<\/p>\n<div id=\"fs-id2628826\">\n<div id=\"fs-id2628827\">\n<p id=\"fs-id2628828\">[latex]{x}^{\\frac{2}{3}}=16[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1973815\">\n<div id=\"fs-id1973816\">\n<p id=\"fs-id1973817\">[latex]{x}^{\\frac{3}{4}}=27[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2426775\">[latex]x=81[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2432476\">\n<div id=\"fs-id2432477\">\n<p id=\"fs-id2432478\">[latex]2{x}^{\\frac{1}{2}}-{x}^{\\frac{1}{4}}=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id3040226\">\n<div id=\"fs-id3040227\">\n<p id=\"fs-id1296521\">[latex]{\\left(x-1\\right)}^{\\frac{3}{4}}=8[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1934614\">[latex]x=17[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id3060328\">\n<div id=\"fs-id3060329\">\n<p id=\"fs-id3060330\">[latex]{\\left(x+1\\right)}^{\\frac{2}{3}}=4[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1763592\">\n<div id=\"fs-id1763593\">\n<p id=\"fs-id1763594\">[latex]{x}^{\\frac{2}{3}}-5{x}^{\\frac{1}{3}}+6=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1239690\">[latex]x=8, x=27[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1709842\">\n<div id=\"fs-id1709843\">\n<p id=\"fs-id1709844\">[latex]{x}^{\\frac{7}{3}}-3{x}^{\\frac{4}{3}}-4{x}^{\\frac{1}{3}}=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id2382870\">For the following exercises, solve the following polynomial equations by grouping and factoring.<\/p>\n<div id=\"fs-id2382874\">\n<div id=\"fs-id2382875\">\n<p id=\"fs-id2382876\">[latex]{x}^{3}+2{x}^{2}-x-2=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1396320\">[latex]x=-2,1,-1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id3111192\">\n<div id=\"fs-id3111193\">\n<p id=\"fs-id3111194\">[latex]3{x}^{3}-6{x}^{2}-27x+54=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1393262\">\n<div id=\"fs-id1393263\">\n<p id=\"fs-id1393264\">[latex]4{y}^{3}-9y=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1567745\">[latex]y=0, \\frac{3}{2}, \\frac{-3}{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2529330\">\n<div id=\"fs-id2529331\">\n<p id=\"fs-id2529332\">[latex]{x}^{3}+3{x}^{2}-25x-75=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1582399\">\n<div id=\"fs-id1582400\">\n<p id=\"fs-id1582402\">[latex]{m}^{3}+{m}^{2}-m-1=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2645204\">[latex]m=1,-1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id586482\">\n<div id=\"fs-id586483\">\n<p id=\"fs-id2998491\">[latex]2{x}^{5}-14{x}^{3}=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2643100\">\n<div id=\"fs-id2643101\">\n<p id=\"fs-id2643102\">[latex]5{x}^{3}+45x=2{x}^{2}+18[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1337966\">[latex]x=\\frac{2}{5},\u00b13i[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id2917413\">For the following exercises, solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions.<\/p>\n<div id=\"fs-id2917418\">\n<div id=\"fs-id2294086\">\n<p id=\"fs-id2294087\">[latex]\\sqrt{3x-1}-2=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2332670\">\n<div id=\"fs-id2332671\">\n<p id=\"fs-id2332672\">[latex]\\sqrt{x-7}=5[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2018882\">[latex]x=32[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2375964\">\n<div id=\"fs-id2375966\">\n<p id=\"fs-id2375967\">[latex]\\sqrt{x-1}=x-7[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2486495\">\n<div id=\"fs-id2486496\">\n<p id=\"fs-id2486497\">[latex]\\sqrt{3t+5}=7[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2385196\">[latex]t=\\frac{44}{3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1551969\">\n<div id=\"fs-id1551970\">\n<p id=\"fs-id1551971\">[latex]\\sqrt{t+1}+9=7[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id3256844\">\n<div id=\"fs-id3256845\">\n<p id=\"fs-id3256846\">[latex]\\sqrt{12-x}=x[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1333069\">[latex]x=3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1357710\">\n<div id=\"fs-id1357711\">\n<p id=\"fs-id1357712\">[latex]\\sqrt{2x+3}-\\sqrt{x+2}=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1441558\">\n<div id=\"fs-id1441559\">\n<p id=\"fs-id1441560\">[latex]\\sqrt{3x+7}+\\sqrt{x+2}=1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2383872\">[latex]x=-2[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1951699\">\n<p id=\"fs-id1951700\">[latex]\\sqrt{2x+3}-\\sqrt{x+1}=1[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1786258\">For the following exercises, solve the equation involving absolute value.<\/p>\n<div id=\"fs-id1150948\">\n<div id=\"fs-id1150949\">\n<p id=\"fs-id1150950\">[latex]|3x-4|=8[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2718106\">[latex]x=4,\\frac{-4}{3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2307448\">\n<div id=\"fs-id2307449\">\n<p id=\"fs-id2307450\">[latex]|2x-3|=-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2410873\">\n<div id=\"fs-id2410874\">\n<p id=\"fs-id2410875\">[latex]|1-4x|-1=5[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2818662\">[latex]x=\\frac{-5}{4},\\frac{7}{4}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2428664\">\n<div id=\"fs-id2428665\">\n<p id=\"fs-id2428666\">[latex]|4x+1|-3=6[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1717824\">\n<div id=\"fs-id1717825\">\n<p id=\"fs-id1717826\">[latex]|2x-1|-7=-2[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1780317\">[latex]x=3,-2[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id3165049\">\n<div id=\"fs-id3165050\">\n<p id=\"fs-id3165051\">[latex]|2x+1|-2=-3[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2382990\">\n<div id=\"fs-id2382991\">\n<p id=\"fs-id2382992\">[latex]|x+5|=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2390987\">[latex]x=-5[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2643356\">\n<div id=\"fs-id2643357\">\n<p id=\"fs-id2643358\">[latex]-|2x+1|=-3[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id2665057\">For the following exercises, solve the equation by identifying the quadratic form. Use a substitute variable and find all real solutions by factoring.<\/p>\n<div id=\"fs-id2665062\">\n<div id=\"fs-id2665063\">\n<p id=\"fs-id2386875\">[latex]{x}^{4}-10{x}^{2}+9=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id3126793\">[latex]x=1,-1,3,-3[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1715518\">\n<div id=\"fs-id1715519\">\n<p id=\"fs-id1715520\">[latex]4{\\left(t-1\\right)}^{2}-9\\left(t-1\\right)=-2[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1577029\">\n<div id=\"fs-id2029121\">\n<p id=\"fs-id2029122\">[latex]{\\left({x}^{2}-1\\right)}^{2}+\\left({x}^{2}-1\\right)-12=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id3042785\">[latex]x=2,-2[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1418519\">\n<div id=\"fs-id1418520\">\n<p id=\"fs-id1418521\">[latex]{\\left(x+1\\right)}^{2}-8\\left(x+1\\right)-9=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2797000\">\n<div id=\"fs-id2797001\">\n<p id=\"fs-id2797002\">[latex]{\\left(x-3\\right)}^{2}-4=0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1518893\">[latex]x=1,5[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2390141\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<p id=\"fs-id2390146\">For the following exercises, solve for the unknown variable.<\/p>\n<div>\n<div>\n<p id=\"fs-id2390152\">[latex]{x}^{-2}-{x}^{-1}-12=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1726737\">\n<div id=\"fs-id1726738\">\n<p id=\"fs-id1726739\">[latex]\\sqrt{{|x|}^{2}}=x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2412232\">All real numbers<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2412235\">\n<div id=\"fs-id2412236\">\n<p id=\"fs-id2412237\">[latex]{t}^{10}-{t}^{5}+1=0[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1276816\">\n<div id=\"fs-id1276817\">\n<p id=\"fs-id1276818\">[latex]|{x}^{2}+2x-36|=12[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2699615\">[latex]x=4,6,-6,-8[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1976510\" class=\"bc-section section\">\n<h4>Real-World Applications<\/h4>\n<p id=\"fs-id1976516\">For the following exercises, use the model for the period of a pendulum,[latex]\\,T,[\/latex]such that[latex]\\,T=2\\pi \\sqrt{\\frac{L}{g}},[\/latex]where the length of the pendulum is <em>L<\/em> and the acceleration due to gravity is[latex]\\,g.[\/latex]<\/p>\n<div id=\"fs-id2500088\">\n<div id=\"fs-id2500089\">\n<p id=\"fs-id2500090\">If the acceleration due to gravity is 9.8 m\/s<sup>2<\/sup> and the period equals 1 s, find the length to the nearest cm (100 cm = 1 m).<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2947584\">\n<div id=\"fs-id2947585\">\n<p id=\"fs-id1762772\">If the gravity is 32 ft\/s<sup>2<\/sup> and the period equals 1 s, find the length to the nearest in. (12 in. = 1 ft). Round your answer to the nearest in.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id3207106\">10 in.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1591913\">For the following exercises, use a model for body surface area, BSA, such that[latex]\\,BSA=\\sqrt{\\frac{wh}{3600}},[\/latex]where <em>w<\/em> = weight in kg and <em>h<\/em> = height in cm.<\/p>\n<div>\n<div id=\"fs-id3070468\">\n<p id=\"fs-id3070470\">Find the height of a 72-kg female to the nearest cm whose[latex]\\,BSA=1.8.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1591172\">\n<div id=\"fs-id1591174\">\n<p id=\"fs-id1591175\">Find the weight of a 177-cm male to the nearest kg whose[latex]\\,BSA=2.1.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2290740\">90 kg<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id2290748\">\n<dt>absolute value equation<\/dt>\n<dd id=\"fs-id2290751\">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-id1333937\">\n<dt>equations in quadratic form<\/dt>\n<dd id=\"fs-id1333940\">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-id1333945\">\n<dt>extraneous solutions<\/dt>\n<dd id=\"fs-id1333948\">any solutions obtained that are not valid in the original equation<\/dd>\n<\/dl>\n<dl id=\"fs-id1547816\">\n<dt>polynomial equation<\/dt>\n<dd id=\"fs-id1547819\">an equation containing a string of terms including numerical coefficients and variables raised to whole-number exponents<\/dd>\n<\/dl>\n<dl id=\"fs-id1547823\">\n<dt>radical equation<\/dt>\n<dd id=\"fs-id1547827\">an equation containing at least one radical term where the variable is part of the radicand<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":291,"menu_order":7,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-46","chapter","type-chapter","status-publish","hentry"],"part":33,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/46","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/users\/291"}],"version-history":[{"count":1,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/46\/revisions"}],"predecessor-version":[{"id":47,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/46\/revisions\/47"}],"part":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/parts\/33"}],"metadata":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/46\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=46"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=46"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=46"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=46"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}