{"id":1824,"date":"2021-09-10T15:45:21","date_gmt":"2021-09-10T15:45:21","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&p=1824"},"modified":"2022-02-07T18:47:12","modified_gmt":"2022-02-07T18:47:12","slug":"solving-quadratic-equations-by-the-square-root-property","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/solving-quadratic-equations-by-the-square-root-property\/","title":{"raw":"Solving Quadratic Equations by the Square Root Property","rendered":"Solving Quadratic Equations by the Square Root Property"},"content":{"raw":"
\r\n

Learning Outcomes<\/h3>\r\n
\r\n
    \r\n \t
  • Use the square root property to solve a quadratic equation<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\nThe\u00a0principal square root<\/strong> of a nonnegative number [latex]x[\/latex] is defined to be the nonnegative number [latex]a[\/latex] such that [latex]a^2=x[\/latex]. We write\r\n

    [latex]\\sqrt{x} = a [\/latex] if [latex]a^{2} = x[\/latex] where [latex]x \\geq 0, a \\geq0[\/latex].<\/p>\r\nFor example, [latex]\\sqrt{9} = 3[\/latex] since [latex]3^{2}=9[\/latex].\r\n\r\nA quadratic equation<\/strong> is an equation which can be written in the form\r\n\r\n[latex]ax^2 +bx +c =0[\/latex] where [latex]a,b,[\/latex] and [latex]c[\/latex] are constants, and [latex]a \\neq 0[\/latex].\r\n\r\nQuadratic equations which can be written in the form [latex]X^2 =k[\/latex], where [latex]X[\/latex]\u00a0is any variable expression and [latex]k[\/latex] is a nonnegative number can be solved by the square root property<\/strong>.\r\n

    \r\n

    The Square Root Property<\/h3>\r\nIf [latex]x^{2}=a[\/latex], then [latex] x=\\sqrt{a}[\/latex] or [latex] -\\sqrt{a}[\/latex].\r\n\r\n \r\n\r\nThe solutions of [latex]x^2=a[\/latex] are called the square roots of a<\/em>.\r\n
      \r\n \t
    • When a<\/em> is positive, [latex]a > 0[\/latex], [latex]x^2=a[\/latex] has two solutions, [latex]+\\sqrt{a},-\\sqrt{a}[\/latex]. [latex]+\\sqrt{a}[\/latex] is the nonnegative square root of a<\/em>, and [latex]-\\sqrt{a}[\/latex] is the negative square root of a<\/em>.<\/li>\r\n \t
    • When a<\/em> is negative, [latex]a < 0[\/latex], [latex]x^2=a[\/latex] has no solutions.<\/li>\r\n \t
    • When a<\/em> is zero, [latex]a = 0[\/latex], [latex]x^2=a[\/latex] has one solution: [latex]a = 0[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n
      \r\n

      Example<\/h3>\r\nSolve using the square root property. [latex]x^{2}=16[\/latex]\r\n\r\n[reveal-answer q=\"793132\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"793132\"]\r\n\r\nBy the square root property,\r\n

      [latex]x = \\pm \\sqrt{16} = \\pm 4[\/latex]<\/p>\r\n

      Then the equation has two solutions, [latex]x=4[\/latex] and [latex]x=-4[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the example above, you can take the square root of both sides easily because there is only one term on each side. In some equations, you may need to isolate the second-degree (squared) expression before applying the square root property.\r\n\r\nIn our first video, we will show more examples of using the square root property to solve a quadratic equation.\r\n\r\nhttps:\/\/youtu.be\/Fj-BP7uaWrI\r\n

      \r\n

      Example<\/h3>\r\nSolve [latex]3x^2-1=74[\/latex].\r\n\r\n[reveal-answer q=\"116387\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"116387\"]\r\n\r\nFirst, isolate [latex]x^2[\/latex].\r\n

      [latex]3x^2-1=74[\/latex]<\/p>\r\nAdd [latex]1[\/latex] to each side.\r\n

      [latex]3x^2=75[\/latex]<\/p>\r\nDivide each side by [latex]3[\/latex].\r\n

      [latex]\\frac{3x^2}{3} = \\frac{75}{3}[\/latex]<\/p>\r\nSimplify.\r\n

      [latex]x^2=25[\/latex]<\/p>\r\nApply the square root property.\r\n

      [latex]x = \\pm \\sqrt{25} = \\pm 5[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n

      \r\n

      Try It<\/h3>\r\n[ohm_question]197330[\/ohm_question]\r\n\r\n<\/div>\r\nSometimes more than just a single variable is being squared.\r\n
      \r\n

      Example<\/h3>\r\nSolve [latex](x-1)^2=16[\/latex].\r\n\r\n[reveal-answer q=\"3817\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"3817\"]\r\n\r\nSince the equation has the second-degree expression isolated, we can begin by applying the\u00a0square root property.\r\n

      [latex]x-1= \\pm \\sqrt{16}[\/latex]<\/p>\r\n

      [latex]x-1= \\pm4[\/latex]<\/p>\r\nThis gives two equations:\r\n

      [latex]x-1=4[\/latex] or [latex]x-1=-4[\/latex]<\/p>\r\nWe solve each equation by adding [latex]1[\/latex] to each side.\r\n

      [latex]x=5[\/latex] or [latex]x=-3[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the next video, you will see more examples of using square roots to solve quadratic equations.\r\n\r\nhttps:\/\/youtu.be\/4H5qZ_-8YM4","rendered":"

      \n

      Learning Outcomes<\/h3>\n
      \n
        \n
      • Use the square root property to solve a quadratic equation<\/li>\n<\/ul>\n<\/section>\n<\/div>\n

        The\u00a0principal square root<\/strong> of a nonnegative number [latex]x[\/latex] is defined to be the nonnegative number [latex]a[\/latex] such that [latex]a^2=x[\/latex]. We write<\/p>\n

        [latex]\\sqrt{x} = a[\/latex] if [latex]a^{2} = x[\/latex] where [latex]x \\geq 0, a \\geq0[\/latex].<\/p>\n

        For example, [latex]\\sqrt{9} = 3[\/latex] since [latex]3^{2}=9[\/latex].<\/p>\n

        A quadratic equation<\/strong> is an equation which can be written in the form<\/p>\n

        [latex]ax^2 +bx +c =0[\/latex] where [latex]a,b,[\/latex] and [latex]c[\/latex] are constants, and [latex]a \\neq 0[\/latex].<\/p>\n

        Quadratic equations which can be written in the form [latex]X^2 =k[\/latex], where [latex]X[\/latex]\u00a0is any variable expression and [latex]k[\/latex] is a nonnegative number can be solved by the square root property<\/strong>.<\/p>\n

        \n

        The Square Root Property<\/h3>\n

        If [latex]x^{2}=a[\/latex], then [latex]x=\\sqrt{a}[\/latex] or [latex]-\\sqrt{a}[\/latex].<\/p>\n

         <\/p>\n

        The solutions of [latex]x^2=a[\/latex] are called the square roots of a<\/em>.<\/p>\n

          \n
        • When a<\/em> is positive, [latex]a > 0[\/latex], [latex]x^2=a[\/latex] has two solutions, [latex]+\\sqrt{a},-\\sqrt{a}[\/latex]. [latex]+\\sqrt{a}[\/latex] is the nonnegative square root of a<\/em>, and [latex]-\\sqrt{a}[\/latex] is the negative square root of a<\/em>.<\/li>\n
        • When a<\/em> is negative, [latex]a < 0[\/latex], [latex]x^2=a[\/latex] has no solutions.<\/li>\n
        • When a<\/em> is zero, [latex]a = 0[\/latex], [latex]x^2=a[\/latex] has one solution: [latex]a = 0[\/latex]<\/li>\n<\/ul>\n<\/div>\n
          \n

          Example<\/h3>\n

          Solve using the square root property. [latex]x^{2}=16[\/latex]<\/p>\n

          Show Solution<\/span><\/p>\n
          \n

          By the square root property,<\/p>\n

          [latex]x = \\pm \\sqrt{16} = \\pm 4[\/latex]<\/p>\n

          Then the equation has two solutions, [latex]x=4[\/latex] and [latex]x=-4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n

          In the example above, you can take the square root of both sides easily because there is only one term on each side. In some equations, you may need to isolate the second-degree (squared) expression before applying the square root property.<\/p>\n

          In our first video, we will show more examples of using the square root property to solve a quadratic equation.<\/p>\n