{"id":15389,"date":"2021-10-11T23:05:42","date_gmt":"2021-10-11T23:05:42","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/read-quadratic-equations-with-complex-solutions\/"},"modified":"2021-10-22T00:33:25","modified_gmt":"2021-10-22T00:33:25","slug":"quadratic-equations-with-complex-solutions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/quadratic-equations-with-complex-solutions\/","title":{"raw":"Quadratic Equations With Complex Solutions","rendered":"Quadratic Equations With Complex Solutions"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use the quadratic formula to solve quadratic equations with complex solutions<\/li>\r\n<\/ul>\r\n<\/div>\r\nWe have seen two outcomes for solutions to\u00a0quadratic equations; either there was one or two real number solutions. We have also learned that it is possible to take the square root of a negative number by using imaginary numbers. Having this new knowledge allows us to explore one more possible outcome when we solve quadratic equations. Consider this equation:\r\n<p style=\"text-align: center;\">[latex]2x^2+3x+6=0[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Using the Quadratic Formula to solve this equation, we first identify a, b, and c.<\/p>\r\n<p style=\"text-align: center;\">[latex]a = 2,b = 3,c = 6[\/latex]<\/p>\r\n<p style=\"text-align: left;\">We can place a, b and c into the quadratic formula and simplify to get the following result:<\/p>\r\n<p style=\"text-align: center;\">[latex]x=-\\frac{3}{4}+\\frac{\\sqrt{-39}}{4}, x=-\\frac{3}{4}-\\frac{\\sqrt{-39}}{4}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Up to this point, we would have said\u00a0that [latex]\\sqrt{-39}[\/latex] is not defined for real numbers and determine that this equation has no solutions. \u00a0But, now that we have defined the square root of a negative number, we can also define a solution to this equation as follows.<\/p>\r\n<p style=\"text-align: center;\">[latex]x=-\\frac{3}{4}+i\\frac{\\sqrt{39}}{4}, x=-\\frac{3}{4}-i\\frac{\\sqrt{39}}{4}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">In this section we will practice simplifying and writing solutions to quadratic equations that are complex. \u00a0We will then present a technique for classifying whether the solution(s) to a quadratic equation will be complex, and how many solutions there will be.<\/p>\r\n<p style=\"text-align: left;\">In our first example, we will work through the process of solving\u00a0a quadratic equation with\u00a0complex solutions. Take note that we will be simplifying complex numbers, so <a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/16-4-1-complex-numbers\/\">if you need a review of how to rewrite the square root of a negative number as an imaginary number, now is a good time<\/a>.<\/p>\r\n\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nUse the Quadratic Formula to solve the equation [latex]x^{2}+2x=-5[\/latex].\r\n\r\n[reveal-answer q=\"654640\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"654640\"]\r\n\r\nFirst write the equation in standard form.\r\n\r\n[latex]\\begin{array}{l}x^{2}+2x=-5\\\\x^{2}+2x+5=0\\,\\,\\,\\,\\,\\\\\\\\a=1,b=2,c=5\\,\\,\\,\\end{array}[\/latex]\r\n\r\n[latex] \\begin{array}{r}{{x}^{2}}\\,\\,\\,+\\,\\,\\,2x\\,\\,\\,+\\,\\,\\,5\\,\\,\\,=\\,\\,\\,0\\\\\\downarrow\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\downarrow\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\downarrow\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\a{{x}^{2}}\\,\\,\\,+\\,\\,\\,bx\\,\\,\\,+\\,\\,\\,c\\,\\,\\,=\\,\\,\\,0\\end{array}[\/latex]\r\n\r\nSubstitute the values into the Quadratic Formula.\r\n\r\n[latex] x=\\frac{-b\\pm \\sqrt{{{b}^{2}}-4ac}}{2a}\\\\x=\\frac{-2\\pm \\sqrt{{{(2)}^{2}}-4(1)(5)}}{2(1)}[\/latex]\r\n\r\nSimplify, being careful to get the signs correct.\r\n\r\n[latex] x=\\frac{-2\\pm \\sqrt{4-20}}{2}[\/latex]\r\n\r\nSimplify some more.\r\n\r\n[latex] x=\\frac{-2\\pm \\sqrt{-16}}{2}[\/latex]\r\n\r\nSimplify the radical, but notice that the number under the radical symbol is negative! The square root of [latex]\u221216[\/latex] is imaginary. [latex] \\sqrt{-16}=4i[\/latex].\r\n\r\n[latex] x=\\frac{-2\\pm 4i}{2}[\/latex]\r\n\r\nSeparate and simplify to find the solutions to the quadratic equation.\r\n\r\n[latex]\\begin{array}{c}x=\\frac{-2+4i}{2}=\\frac{-2}{2}+\\frac{4i}{2}=-1+2i\\\\\\\\\\text{or}\\\\\\\\x=\\frac{-2-4i}{2}=\\frac{-2}{2}-\\frac{2i}{4}\\cdot \\frac{2}{2}=-1-2i\\end{array}[\/latex]\r\n\r\nTherefore, [latex]x=-1+2i[\/latex] or [latex]-1-2i[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWe can check these solutions in the original equation. Be careful when you expand the squares, and replace [latex]i^{2}[\/latex]\u00a0with [latex]-1[\/latex].\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\begin{array}{r}x=-1+2i\\\\x^{2}+2x=-5\\\\\\left(-1+2i\\right)^{2}+2\\left(-1+2i\\right)=-5\\\\1-4i+4i^{2}-2+4i=-5\\\\1-4i+4\\left(-1\\right)-2+4i=-5\\\\1-4-2=-5\\\\-5=-5\\end{array}[\/latex]<\/td>\r\n<td>[latex]\\begin{array}{r}x=-1-2i\\\\x^{2}+2x=-5\\\\\\left(-1-2i\\right)^{2}+2\\left(-1-2i\\right)=-5\\\\1+4i+4i^{2}-2-4i=-5\\\\1+4i+4\\left(-1\\right)-2-4i=-5\\\\1-4-2=-5\\\\-5=-5\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<p id=\"fs-id1165135500790\">Use the quadratic formula to solve [latex]{x}^{2}+x+2=0[\/latex].<\/p>\r\n[reveal-answer q=\"144216\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"144216\"]\r\n\r\nFirst, we identify the coefficients: [latex]a=1,b=1[\/latex], and [latex]c=2[\/latex]. Substitute these values into the quadratic formula. [latex]\\begin{array}{l}x\\hfill&amp;=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}\\hfill \\\\\\hfill&amp;=\\frac{-\\left(1\\right)\\pm \\sqrt{{\\left(1\\right)}^{2}-\\left(4\\right)\\cdot \\left(1\\right)\\cdot \\left(2\\right)}}{2\\cdot 1}\\hfill \\\\\\hfill&amp;=\\frac{-1\\pm \\sqrt{1 - 8}}{2}\\hfill \\\\ \\hfill&amp;=\\frac{-1\\pm \\sqrt{-7}}{2}\\hfill \\\\\\hfill&amp;=\\frac{-1\\pm i\\sqrt{7}}{2}\\hfill \\end{array}[\/latex]\r\n\r\nNow we can separate the expression [latex]\\frac{-1\\pm i\\sqrt{7}}{2}[\/latex] into two solutions:\r\n\r\n[latex]-\\frac{1}{2}+\\frac{ i\\sqrt{7}}{2}[\/latex]\r\n\r\n[latex]-\\frac{1}{2}-\\frac{ i\\sqrt{7}}{2}[\/latex]\r\n\r\n&nbsp;\r\n\r\nThe solutions to the equation are [latex]x=\\frac{-1}{2}+\\frac{i\\sqrt{7}}{2}[\/latex] and [latex]x=\\frac{-1}{2}-\\frac{i\\sqrt{7}}{2}[\/latex]. It is important that you separate the real and imaginary part as this is proper complex number form.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]25644[\/ohm_question]\r\n\r\n<\/div>\r\nThe following video gives another example of how to use the quadratic formula to find solutions to a quadratic equation that has complex solutions.\r\n\r\nhttps:\/\/youtu.be\/11EwTcRMPn8\r\n<h2>Summary<\/h2>\r\nQuadratic equations can have complex solutions.\r\n<h2>Contribute!<\/h2>\r\n<div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div>\r\n<a style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\" href=\"https:\/\/docs.google.com\/document\/d\/1V3MCUDZyeEckXEznD15tebiKDZUmaAuxOurtL0tqeOc\" target=\"_blank\" rel=\"noopener\">Improve this page<\/a><a style=\"margin-left: 16px;\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\" target=\"_blank\" rel=\"noopener\">Learn More<\/a>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use the quadratic formula to solve quadratic equations with complex solutions<\/li>\n<\/ul>\n<\/div>\n<p>We have seen two outcomes for solutions to\u00a0quadratic equations; either there was one or two real number solutions. We have also learned that it is possible to take the square root of a negative number by using imaginary numbers. Having this new knowledge allows us to explore one more possible outcome when we solve quadratic equations. Consider this equation:<\/p>\n<p style=\"text-align: center;\">[latex]2x^2+3x+6=0[\/latex]<\/p>\n<p style=\"text-align: left;\">Using the Quadratic Formula to solve this equation, we first identify a, b, and c.<\/p>\n<p style=\"text-align: center;\">[latex]a = 2,b = 3,c = 6[\/latex]<\/p>\n<p style=\"text-align: left;\">We can place a, b and c into the quadratic formula and simplify to get the following result:<\/p>\n<p style=\"text-align: center;\">[latex]x=-\\frac{3}{4}+\\frac{\\sqrt{-39}}{4}, x=-\\frac{3}{4}-\\frac{\\sqrt{-39}}{4}[\/latex]<\/p>\n<p style=\"text-align: left;\">Up to this point, we would have said\u00a0that [latex]\\sqrt{-39}[\/latex] is not defined for real numbers and determine that this equation has no solutions. \u00a0But, now that we have defined the square root of a negative number, we can also define a solution to this equation as follows.<\/p>\n<p style=\"text-align: center;\">[latex]x=-\\frac{3}{4}+i\\frac{\\sqrt{39}}{4}, x=-\\frac{3}{4}-i\\frac{\\sqrt{39}}{4}[\/latex]<\/p>\n<p style=\"text-align: left;\">In this section we will practice simplifying and writing solutions to quadratic equations that are complex. \u00a0We will then present a technique for classifying whether the solution(s) to a quadratic equation will be complex, and how many solutions there will be.<\/p>\n<p style=\"text-align: left;\">In our first example, we will work through the process of solving\u00a0a quadratic equation with\u00a0complex solutions. Take note that we will be simplifying complex numbers, so <a href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/16-4-1-complex-numbers\/\">if you need a review of how to rewrite the square root of a negative number as an imaginary number, now is a good time<\/a>.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Use the Quadratic Formula to solve the equation [latex]x^{2}+2x=-5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q654640\">Show Solution<\/span><\/p>\n<div id=\"q654640\" class=\"hidden-answer\" style=\"display: none\">\n<p>First write the equation in standard form.<\/p>\n<p>[latex]\\begin{array}{l}x^{2}+2x=-5\\\\x^{2}+2x+5=0\\,\\,\\,\\,\\,\\\\\\\\a=1,b=2,c=5\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>[latex]\\begin{array}{r}{{x}^{2}}\\,\\,\\,+\\,\\,\\,2x\\,\\,\\,+\\,\\,\\,5\\,\\,\\,=\\,\\,\\,0\\\\\\downarrow\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\downarrow\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\downarrow\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\a{{x}^{2}}\\,\\,\\,+\\,\\,\\,bx\\,\\,\\,+\\,\\,\\,c\\,\\,\\,=\\,\\,\\,0\\end{array}[\/latex]<\/p>\n<p>Substitute the values into the Quadratic Formula.<\/p>\n<p>[latex]x=\\frac{-b\\pm \\sqrt{{{b}^{2}}-4ac}}{2a}\\\\x=\\frac{-2\\pm \\sqrt{{{(2)}^{2}}-4(1)(5)}}{2(1)}[\/latex]<\/p>\n<p>Simplify, being careful to get the signs correct.<\/p>\n<p>[latex]x=\\frac{-2\\pm \\sqrt{4-20}}{2}[\/latex]<\/p>\n<p>Simplify some more.<\/p>\n<p>[latex]x=\\frac{-2\\pm \\sqrt{-16}}{2}[\/latex]<\/p>\n<p>Simplify the radical, but notice that the number under the radical symbol is negative! The square root of [latex]\u221216[\/latex] is imaginary. [latex]\\sqrt{-16}=4i[\/latex].<\/p>\n<p>[latex]x=\\frac{-2\\pm 4i}{2}[\/latex]<\/p>\n<p>Separate and simplify to find the solutions to the quadratic equation.<\/p>\n<p>[latex]\\begin{array}{c}x=\\frac{-2+4i}{2}=\\frac{-2}{2}+\\frac{4i}{2}=-1+2i\\\\\\\\\\text{or}\\\\\\\\x=\\frac{-2-4i}{2}=\\frac{-2}{2}-\\frac{2i}{4}\\cdot \\frac{2}{2}=-1-2i\\end{array}[\/latex]<\/p>\n<p>Therefore, [latex]x=-1+2i[\/latex] or [latex]-1-2i[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>We can check these solutions in the original equation. Be careful when you expand the squares, and replace [latex]i^{2}[\/latex]\u00a0with [latex]-1[\/latex].<\/p>\n<table>\n<tbody>\n<tr>\n<td>[latex]\\begin{array}{r}x=-1+2i\\\\x^{2}+2x=-5\\\\\\left(-1+2i\\right)^{2}+2\\left(-1+2i\\right)=-5\\\\1-4i+4i^{2}-2+4i=-5\\\\1-4i+4\\left(-1\\right)-2+4i=-5\\\\1-4-2=-5\\\\-5=-5\\end{array}[\/latex]<\/td>\n<td>[latex]\\begin{array}{r}x=-1-2i\\\\x^{2}+2x=-5\\\\\\left(-1-2i\\right)^{2}+2\\left(-1-2i\\right)=-5\\\\1+4i+4i^{2}-2-4i=-5\\\\1+4i+4\\left(-1\\right)-2-4i=-5\\\\1-4-2=-5\\\\-5=-5\\end{array}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p id=\"fs-id1165135500790\">Use the quadratic formula to solve [latex]{x}^{2}+x+2=0[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q144216\">Show Solution<\/span><\/p>\n<div id=\"q144216\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, we identify the coefficients: [latex]a=1,b=1[\/latex], and [latex]c=2[\/latex]. Substitute these values into the quadratic formula. [latex]\\begin{array}{l}x\\hfill&=\\frac{-b\\pm \\sqrt{{b}^{2}-4ac}}{2a}\\hfill \\\\\\hfill&=\\frac{-\\left(1\\right)\\pm \\sqrt{{\\left(1\\right)}^{2}-\\left(4\\right)\\cdot \\left(1\\right)\\cdot \\left(2\\right)}}{2\\cdot 1}\\hfill \\\\\\hfill&=\\frac{-1\\pm \\sqrt{1 - 8}}{2}\\hfill \\\\ \\hfill&=\\frac{-1\\pm \\sqrt{-7}}{2}\\hfill \\\\\\hfill&=\\frac{-1\\pm i\\sqrt{7}}{2}\\hfill \\end{array}[\/latex]<\/p>\n<p>Now we can separate the expression [latex]\\frac{-1\\pm i\\sqrt{7}}{2}[\/latex] into two solutions:<\/p>\n<p>[latex]-\\frac{1}{2}+\\frac{ i\\sqrt{7}}{2}[\/latex]<\/p>\n<p>[latex]-\\frac{1}{2}-\\frac{ i\\sqrt{7}}{2}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<p>The solutions to the equation are [latex]x=\\frac{-1}{2}+\\frac{i\\sqrt{7}}{2}[\/latex] and [latex]x=\\frac{-1}{2}-\\frac{i\\sqrt{7}}{2}[\/latex]. It is important that you separate the real and imaginary part as this is proper complex number form.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm25644\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=25644&theme=oea&iframe_resize_id=ohm25644&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The following video gives another example of how to use the quadratic formula to find solutions to a quadratic equation that has complex solutions.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Ex:  Quadratic Formula - Complex Solutions\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/11EwTcRMPn8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Summary<\/h2>\n<p>Quadratic equations can have complex solutions.<\/p>\n<h2>Contribute!<\/h2>\n<div style=\"margin-bottom: 8px;\">Did you have an idea for improving this content? We\u2019d love your input.<\/div>\n<p><a style=\"font-size: 10pt; font-weight: 600; color: #077fab; text-decoration: none; border: 2px solid #077fab; border-radius: 7px; padding: 5px 25px; text-align: center; cursor: pointer; line-height: 1.5em;\" href=\"https:\/\/docs.google.com\/document\/d\/1V3MCUDZyeEckXEznD15tebiKDZUmaAuxOurtL0tqeOc\" target=\"_blank\" rel=\"noopener\">Improve this page<\/a><a style=\"margin-left: 16px;\" href=\"https:\/\/docs.google.com\/document\/d\/1vy-T6DtTF-BbMfpVEI7VP_R7w2A4anzYZLXR8Pk4Fu4\" target=\"_blank\" rel=\"noopener\">Learn More<\/a><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-15389\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex: Quadratic Formula - Complex Solutions. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/11EwTcRMPn8\">https:\/\/youtu.be\/11EwTcRMPn8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":167848,"menu_order":15,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex: Quadratic Formula - Complex Solutions\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/11EwTcRMPn8\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et al.\",\"organization\":\"OpenStax\",\"url\":\" http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"482a5524ebbe4ca596b63be2557597bf, 74424c30f2994d5881b6b2351807af13, ebed27ec79b4433683a3da150ab816fe","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-15389","chapter","type-chapter","status-publish","hentry"],"part":10990,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15389","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/users\/167848"}],"version-history":[{"count":4,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15389\/revisions"}],"predecessor-version":[{"id":15853,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15389\/revisions\/15853"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/parts\/10990"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15389\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/media?parent=15389"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapter-type?post=15389"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/contributor?post=15389"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/license?post=15389"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}