{"id":395,"date":"2015-10-26T17:52:07","date_gmt":"2015-10-26T17:52:07","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=395"},"modified":"2015-11-12T18:38:00","modified_gmt":"2015-11-12T18:38:00","slug":"completing-the-square","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/completing-the-square\/","title":{"raw":"Completing the Square","rendered":"Completing the Square"},"content":{"raw":"Not all quadratic equations can be factored or can be solved in their original form using the square root property. In these cases, we may use a method for solving a quadratic equation<\/strong> known as completing the square<\/strong>. Using this method, we add or subtract terms to both sides of the equation until we have a perfect square trinomial on one side of the equal sign. We then apply the square root property. To complete the square, the leading coefficient, a<\/em>, must equal 1. If it does not, then divide the entire equation by a<\/em>. Then, we can use the following procedures to solve a quadratic equation by completing the square.\r\n\r\nWe will use the example [latex]{x}^{2}+4x+1=0[\/latex] to illustrate each step.\r\n
    \r\n\t
  1. Given a quadratic equation that cannot be factored, and with [latex]a=1[\/latex], first add or subtract the constant term to the right sign of the equal sign.\r\n
    [latex]{x}^{2}+4x=-1[\/latex]<\/div><\/li>\r\n\t
  2. Multiply the b <\/em>term by [latex]\\frac{1}{2}[\/latex] and square it.\r\n
    [latex]\\begin{array}{l}\\frac{1}{2}\\left(4\\right)=2\\hfill \\\\ {2}^{2}=4\\hfill \\end{array}[\/latex]<\/div><\/li>\r\n\t
  3. Add [latex]{\\left(\\frac{1}{2}b\\right)}^{2}[\/latex] to both sides of the equal sign and simplify the right side. We have\r\n
    [latex]\\begin{array}{l}{x}^{2}+4x+4=-1+4\\hfill \\\\ {x}^{2}+4x+4=3\\hfill \\end{array}[\/latex]<\/div><\/li>\r\n\t
  4. The left side of the equation can now be factored as a perfect square.\r\n
    [latex]\\begin{array}{l}{x}^{2}+4x+4=3\\hfill \\\\ {\\left(x+2\\right)}^{2}=3\\hfill \\end{array}[\/latex]<\/div><\/li>\r\n\t
  5. Use the square root property and solve.\r\n
    [latex]\\begin{array}{l}\\sqrt{{\\left(x+2\\right)}^{2}}=\\pm \\sqrt{3}\\hfill \\\\ x+2=\\pm \\sqrt{3}\\hfill \\\\ x=-2\\pm \\sqrt{3}\\hfill \\end{array}[\/latex]<\/div><\/li>\r\n\t
  6. The solutions are [latex]x=-2+\\sqrt{3}[\/latex], [latex]x=-2-\\sqrt{3}[\/latex].<\/li>\r\n<\/ol>\r\n
    \r\n

    Example 8: Solving a Quadratic by Completing the Square<\/h3>\r\nSolve the quadratic equation by completing the square: [latex]{x}^{2}-3x - 5=0[\/latex].\r\n\r\n<\/div>\r\n
    \r\n

    Solution<\/h3>\r\nFirst, move the constant term to the right side of the equal sign.\r\n
    [latex]{x}^{2}-3x=5[\/latex]<\/div>\r\nThen, take [latex]\\frac{1}{2}[\/latex] of the b <\/em>term and square it.\r\n
    [latex]\\begin{array}{l}\\frac{1}{2}\\left(-3\\right)=-\\frac{3}{2}\\hfill \\\\ {\\left(-\\frac{3}{2}\\right)}^{2}=\\frac{9}{4}\\hfill \\end{array}[\/latex]<\/div>\r\nAdd the result to both sides of the equal sign.\r\n
    [latex]\\begin{array}{l}\\text{ }{x}^{2}-3x+{\\left(-\\frac{3}{2}\\right)}^{2}=5+{\\left(-\\frac{3}{2}\\right)}^{2}\\hfill \\\\ {x}^{2}-3x+\\frac{9}{4}=5+\\frac{9}{4}\\hfill \\end{array}[\/latex]<\/div>\r\nFactor the left side as a perfect square and simplify the right side.\r\n
    [latex]{\\left(x-\\frac{3}{2}\\right)}^{2}=\\frac{29}{4}[\/latex]<\/div>\r\nUse the square root property and solve.\r\n
    [latex]\\begin{array}{l}\\sqrt{{\\left(x-\\frac{3}{2}\\right)}^{2}}\\hfill&=\\pm \\sqrt{\\frac{29}{4}}\\hfill \\\\ \\left(x-\\frac{3}{2}\\right)\\hfill&=\\pm \\frac{\\sqrt{29}}{2}\\hfill \\\\ x\\hfill&=\\frac{3}{2}\\pm \\frac{\\sqrt{29}}{2}\\hfill \\end{array}[\/latex]<\/div>\r\nThe solutions are [latex]x=\\frac{3}{2}+\\frac{\\sqrt{29}}{2}[\/latex], [latex]x=\\frac{3}{2}-\\frac{\\sqrt{29}}{2}[\/latex].\r\n\r\n<\/div>\r\n
    \r\n

    Try It 7<\/h3>\r\nSolve by completing the square: [latex]{x}^{2}-6x=13[\/latex].\r\n\r\nSolution<\/a>\r\n\r\n<\/div>","rendered":"

    Not all quadratic equations can be factored or can be solved in their original form using the square root property. In these cases, we may use a method for solving a quadratic equation<\/strong> known as completing the square<\/strong>. Using this method, we add or subtract terms to both sides of the equation until we have a perfect square trinomial on one side of the equal sign. We then apply the square root property. To complete the square, the leading coefficient, a<\/em>, must equal 1. If it does not, then divide the entire equation by a<\/em>. Then, we can use the following procedures to solve a quadratic equation by completing the square.<\/p>\n

    We will use the example [latex]{x}^{2}+4x+1=0[\/latex] to illustrate each step.<\/p>\n

      \n
    1. Given a quadratic equation that cannot be factored, and with [latex]a=1[\/latex], first add or subtract the constant term to the right sign of the equal sign.\n
      [latex]{x}^{2}+4x=-1[\/latex]<\/div>\n<\/li>\n
    2. Multiply the b <\/em>term by [latex]\\frac{1}{2}[\/latex] and square it.\n
      [latex]\\begin{array}{l}\\frac{1}{2}\\left(4\\right)=2\\hfill \\\\ {2}^{2}=4\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n
    3. Add [latex]{\\left(\\frac{1}{2}b\\right)}^{2}[\/latex] to both sides of the equal sign and simplify the right side. We have\n
      [latex]\\begin{array}{l}{x}^{2}+4x+4=-1+4\\hfill \\\\ {x}^{2}+4x+4=3\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n
    4. The left side of the equation can now be factored as a perfect square.\n
      [latex]\\begin{array}{l}{x}^{2}+4x+4=3\\hfill \\\\ {\\left(x+2\\right)}^{2}=3\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n
    5. Use the square root property and solve.\n
      [latex]\\begin{array}{l}\\sqrt{{\\left(x+2\\right)}^{2}}=\\pm \\sqrt{3}\\hfill \\\\ x+2=\\pm \\sqrt{3}\\hfill \\\\ x=-2\\pm \\sqrt{3}\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n
    6. The solutions are [latex]x=-2+\\sqrt{3}[\/latex], [latex]x=-2-\\sqrt{3}[\/latex].<\/li>\n<\/ol>\n
      \n

      Example 8: Solving a Quadratic by Completing the Square<\/h3>\n

      Solve the quadratic equation by completing the square: [latex]{x}^{2}-3x - 5=0[\/latex].<\/p>\n<\/div>\n

      \n

      Solution<\/h3>\n

      First, move the constant term to the right side of the equal sign.<\/p>\n

      [latex]{x}^{2}-3x=5[\/latex]<\/div>\n

      Then, take [latex]\\frac{1}{2}[\/latex] of the b <\/em>term and square it.<\/p>\n

      [latex]\\begin{array}{l}\\frac{1}{2}\\left(-3\\right)=-\\frac{3}{2}\\hfill \\\\ {\\left(-\\frac{3}{2}\\right)}^{2}=\\frac{9}{4}\\hfill \\end{array}[\/latex]<\/div>\n

      Add the result to both sides of the equal sign.<\/p>\n

      [latex]\\begin{array}{l}\\text{ }{x}^{2}-3x+{\\left(-\\frac{3}{2}\\right)}^{2}=5+{\\left(-\\frac{3}{2}\\right)}^{2}\\hfill \\\\ {x}^{2}-3x+\\frac{9}{4}=5+\\frac{9}{4}\\hfill \\end{array}[\/latex]<\/div>\n

      Factor the left side as a perfect square and simplify the right side.<\/p>\n

      [latex]{\\left(x-\\frac{3}{2}\\right)}^{2}=\\frac{29}{4}[\/latex]<\/div>\n

      Use the square root property and solve.<\/p>\n

      [latex]\\begin{array}{l}\\sqrt{{\\left(x-\\frac{3}{2}\\right)}^{2}}\\hfill&=\\pm \\sqrt{\\frac{29}{4}}\\hfill \\\\ \\left(x-\\frac{3}{2}\\right)\\hfill&=\\pm \\frac{\\sqrt{29}}{2}\\hfill \\\\ x\\hfill&=\\frac{3}{2}\\pm \\frac{\\sqrt{29}}{2}\\hfill \\end{array}[\/latex]<\/div>\n

      The solutions are [latex]x=\\frac{3}{2}+\\frac{\\sqrt{29}}{2}[\/latex], [latex]x=\\frac{3}{2}-\\frac{\\sqrt{29}}{2}[\/latex].<\/p>\n<\/div>\n

      \n

      Try It 7<\/h3>\n

      Solve by completing the square: [latex]{x}^{2}-6x=13[\/latex].<\/p>\n

      Solution<\/a><\/p>\n<\/div>\n\n\t\t\t

      \n\t\t\t

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