{"id":1255,"date":"2024-05-07T22:27:36","date_gmt":"2024-05-07T22:27:36","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=1255"},"modified":"2025-08-13T15:40:04","modified_gmt":"2025-08-13T15:40:04","slug":"quadratic-equations-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/quadratic-equations-learn-it-2\/","title":{"raw":"Quadratic Equations: Learn It 2","rendered":"Quadratic Equations: Learn It 2"},"content":{"raw":"

Using the Square Root Property<\/h2>\r\nWhen there is no linear term in the equation, another method of solving a quadratic equation is by using the square root property<\/strong>, in which we isolate the [latex]{x}^{2}[\/latex] term and take the square root of the number on the other side of the equal sign. Keep in mind that sometimes we may have to manipulate the equation to isolate the [latex]{x}^{2}[\/latex] term so that the square root property can be used.\r\n\r\n
\r\n

square root property<\/h3>\r\nWith the [latex]{x}^{2}[\/latex] term isolated, the square root property states that:\r\n
[latex]\\text{if }{x}^{2}=k,\\text{then }x=\\pm \\sqrt{k}[\/latex]<\/div>\r\nwhere [latex]k[\/latex] is a non-negative real number.\r\n\r\n<\/section>
Recall that the\u00a0princicpal square root<\/em> of a number such as [latex]\\sqrt{9}[\/latex] is the non-negative root, [latex]3[\/latex]. Note that there is a difference between [latex]\\sqrt{9}[\/latex]\u00a0 and [latex]{x}^{2}=9[\/latex]. In the case of [latex]{x}^{2}=9[\/latex], we seek all numbers\u00a0<\/em>whose square is [latex]9[\/latex], that is [latex]x=\\pm 3 [\/latex]. This means that if the square of a variable equals a number, then the variable itself is the positive or negative square root of that number.<\/section>
How To: Given a quadratic equation with an [latex]{x}^{2}[\/latex] term but no [latex]x[\/latex] term, use the square root property to solve it<\/strong>\r\n
    \r\n \t
  1. Isolate the [latex]{x}^{2}[\/latex] term on one side of the equal sign.<\/li>\r\n \t
  2. Take the square root of both sides of the equation, putting a [latex]\\pm [\/latex] sign before the expression on the side opposite the squared term.<\/li>\r\n \t
  3. Simplify the numbers on the side with the [latex]\\pm [\/latex] sign.<\/li>\r\n<\/ol>\r\n<\/section>
    Solve the quadratic using the square root property:
    [latex]{x}^{2}=8[\/latex].<\/center>[latex]\\begin{align*} \\text{Given equation:} & \\quad x^2 = 8 \\\\ \\text{Apply the square root property:} & \\quad x = \\pm \\sqrt{8} \\\\ \\text{Simplify the square root:} & \\quad \\sqrt{8} = \\sqrt{4 \\times 2} = 2\\sqrt{2} \\\\ \\text{Thus, the solution is:} & \\quad x = \\pm 2\\sqrt{2} \\end{align*}[\/latex]<\/section>
    Solve the quadratic equation using the square root property:
    [latex]3{\\left(x - 4\\right)}^{2}=15[\/latex]<\/center>[reveal-answer q=\"352957\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"352957\"][latex]\\begin{align*} \\text{Isolate the squared term:} & \\quad (x - 4)^2 = \\frac{15}{3} \\\\ & \\quad (x - 4)^2 = 5 \\\\ \\text{Apply the square root property:} & \\quad x - 4 = \\pm \\sqrt{5} \\\\ \\text{Solve for } x: & \\quad x = 4 \\pm \\sqrt{5} \\\\ & \\quad x = 4 + \\sqrt{5} \\quad \\text{or} \\quad x = 4 - \\sqrt{5} \\end{align*}[\/latex][\/hidden-answer]<\/section>
    Solve the quadratic equation using the square root property:
    [latex]x^2+4=0[\/latex]<\/center>[reveal-answer q=\"82662\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"82662\"]To apply the square root property, isolate the [latex]x^2[\/latex] term on one side: [latex]x^2=-4[\/latex].Applying the square root property: [latex]x = \\pm \\sqrt{-4} [\/latex]Since the square root of a negative number is not defined in the real number system, there are no real solutions to this equation.[\/hidden-answer]<\/section>
    [ohm2_question hide_question_numbers=1]18963[\/ohm2_question]<\/section>\r\n

    Using the Pythagorean Theorem<\/h3>\r\nOne of the most famous formulas in mathematics is the Pythagorean Theorem<\/strong>. It is based on a right triangle and states the relationship among the lengths of the sides as [latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex], where [latex]a[\/latex] and [latex]b[\/latex] refer to the legs of a right triangle adjacent to the [latex]90^\\circ [\/latex] angle, and [latex]c[\/latex] refers to the hypotenuse. It has immeasurable uses in architecture, engineering, the sciences, geometry, trigonometry, and algebra, and in everyday applications.\r\n\r\nWe use the Pythagorean Theorem to solve for the length of one side of a triangle when we have the lengths of the other two. Because each of the terms is squared in the theorem, when we are solving for a side of a triangle, we have a quadratic equation. We can use the methods for solving quadratic equations that we learned in this section to solve for the missing side.\r\n\r\nThe Pythagorean Theorem is given as\r\n
    [latex]{a}^{2}+{b}^{2}={c}^{2}[\/latex]<\/div>\r\nwhere [latex]a[\/latex] and [latex]b[\/latex] refer to the legs of a right triangle adjacent to the [latex]{90}^{\\circ }[\/latex] angle, and [latex]c[\/latex] refers to the hypotenuse.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Right Right triangle with labels[\/caption]\r\n\r\n
    Find the length of the missing side of the right triangle.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Right Right triangle with labels[\/caption]\r\n\r\n[reveal-answer q=\"111347\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"111347\"]As we have measurements for side [latex]b[\/latex] and the hypotenuse, the missing side is [latex]a[\/latex].<\/em>\r\n
    [latex]\\begin{array}{l}{a}^{2}+{b}^{2}={c}^{2}\\hfill \\\\ {a}^{2}+{\\left(4\\right)}^{2}={\\left(12\\right)}^{2}\\hfill \\\\ {a}^{2}+16=144\\hfill \\\\ {a}^{2}=128\\hfill \\\\ a=\\sqrt{128}\\hfill \\\\ a=8\\sqrt{2}\\hfill \\end{array}[\/latex]<\/center>\r\n[\/hidden-answer]<\/section>
    [ohm2_question hide_question_numbers=1]18964[\/ohm2_question]<\/section>","rendered":"

    Using the Square Root Property<\/h2>\n

    When there is no linear term in the equation, another method of solving a quadratic equation is by using the square root property<\/strong>, in which we isolate the [latex]{x}^{2}[\/latex] term and take the square root of the number on the other side of the equal sign. Keep in mind that sometimes we may have to manipulate the equation to isolate the [latex]{x}^{2}[\/latex] term so that the square root property can be used.<\/p>\n

    \n

    square root property<\/h3>\n

    With the [latex]{x}^{2}[\/latex] term isolated, the square root property states that:<\/p>\n

    [latex]\\text{if }{x}^{2}=k,\\text{then }x=\\pm \\sqrt{k}[\/latex]<\/div>\n

    where [latex]k[\/latex] is a non-negative real number.<\/p>\n<\/section>\n

    Recall that the\u00a0princicpal square root<\/em> of a number such as [latex]\\sqrt{9}[\/latex] is the non-negative root, [latex]3[\/latex]. Note that there is a difference between [latex]\\sqrt{9}[\/latex]\u00a0 and [latex]{x}^{2}=9[\/latex]. In the case of [latex]{x}^{2}=9[\/latex], we seek all numbers\u00a0<\/em>whose square is [latex]9[\/latex], that is [latex]x=\\pm 3[\/latex]. This means that if the square of a variable equals a number, then the variable itself is the positive or negative square root of that number.<\/section>\n
    How To: Given a quadratic equation with an [latex]{x}^{2}[\/latex] term but no [latex]x[\/latex] term, use the square root property to solve it<\/strong><\/p>\n
      \n
    1. Isolate the [latex]{x}^{2}[\/latex] term on one side of the equal sign.<\/li>\n
    2. Take the square root of both sides of the equation, putting a [latex]\\pm[\/latex] sign before the expression on the side opposite the squared term.<\/li>\n
    3. Simplify the numbers on the side with the [latex]\\pm[\/latex] sign.<\/li>\n<\/ol>\n<\/section>\n
      Solve the quadratic using the square root property:<\/p>\n
      [latex]{x}^{2}=8[\/latex].<\/div>\n

      [latex]\\begin{align*} \\text{Given equation:} & \\quad x^2 = 8 \\\\ \\text{Apply the square root property:} & \\quad x = \\pm \\sqrt{8} \\\\ \\text{Simplify the square root:} & \\quad \\sqrt{8} = \\sqrt{4 \\times 2} = 2\\sqrt{2} \\\\ \\text{Thus, the solution is:} & \\quad x = \\pm 2\\sqrt{2} \\end{align*}[\/latex]<\/section>\n

      Solve the quadratic equation using the square root property:<\/p>\n
      [latex]3{\\left(x - 4\\right)}^{2}=15[\/latex]<\/div>\n