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Study Guides > MATH 1314: College Algebra

Using the Square Root Property

When there is no linear term in the equation, another method of solving a quadratic equation is by using the square root property, in which we isolate the x2{x}^{2} term and take the square root of the number on the other side of the equals sign. Keep in mind that sometimes we may have to manipulate the equation to isolate the x2{x}^{2} term so that the square root property can be used.

A General Note: The Square Root Property

With the x2{x}^{2} term isolated, the square root property states that:
if x2=k,then x=±k\text{if }{x}^{2}=k,\text{then }x=\pm \sqrt{k}
where k is a nonzero real number.

How To: Given a quadratic equation with an x2{x}^{2} term but no xx term, use the square root property to solve it.

  1. Isolate the x2{x}^{2} term on one side of the equal sign.
  2. Take the square root of both sides of the equation, putting a ±\pm sign before the expression on the side opposite the squared term.
  3. Simplify the numbers on the side with the ±\pm sign.

Example 6: Solving a Simple Quadratic Equation Using the Square Root Property

Solve the quadratic using the square root property: x2=8{x}^{2}=8.

Solution

Take the square root of both sides, and then simplify the radical. Remember to use a pm\\pm sign before the radical symbol.
x2=8x=±8=±22\begin{array}{l}{x}^{2}\hfill&=8\hfill \\ x\hfill&=\pm \sqrt{8}\hfill \\ \hfill&=\pm 2\sqrt{2}\hfill \end{array}
The solutions are x=22x=2\sqrt{2}, x=22x=-2\sqrt{2}.

Example 7: Solving a Quadratic Equation Using the Square Root Property

Solve the quadratic equation: 4x2+1=74{x}^{2}+1=7

Solution

First, isolate the x2{x}^{2} term. Then take the square root of both sides.
4x2+1=74x2=6x2=64x=±62\begin{array}{l}4{x}^{2}+1=7\hfill \\ 4{x}^{2}=6\hfill \\ {x}^{2}=\frac{6}{4}\hfill \\ x=\pm \frac{\sqrt{6}}{2}\hfill \end{array}
The solutions are x=62x=\frac{\sqrt{6}}{2}, x=62x=-\frac{\sqrt{6}}{2}.

Try It 6

Solve the quadratic equation using the square root property: 3(x4)2=153{\left(x - 4\right)}^{2}=15. Solution

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