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 [latex]{x}^{2}[/latex] 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 [latex]{x}^{2}[/latex] term so that the square root property can be used.

A General Note: The Square Root Property

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

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

where k is a nonzero real number.

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.

  1. Isolate the [latex]{x}^{2}[/latex] term on one side of the equal sign.
  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.
  3. Simplify the numbers on the side with the [latex]\pm[/latex] sign.

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

Solve the quadratic using the square root property: [latex]{x}^{2}=8[/latex].

Solution

Take the square root of both sides, and then simplify the radical. Remember to use a [latex]\\pm[/latex] sign before the radical symbol.

[latex]\begin{array}{l}{x}^{2}\hfill&=8\hfill \\ x\hfill&=\pm \sqrt{8}\hfill \\ \hfill&=\pm 2\sqrt{2}\hfill \end{array}[/latex]

The solutions are [latex]x=2\sqrt{2}[/latex], [latex]x=-2\sqrt{2}[/latex].

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

Solve the quadratic equation: [latex]4{x}^{2}+1=7[/latex]

Solution

First, isolate the [latex]{x}^{2}[/latex] term. Then take the square root of both sides.

[latex]\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}[/latex]

The solutions are [latex]x=\frac{\sqrt{6}}{2}[/latex], [latex]x=-\frac{\sqrt{6}}{2}[/latex].

Try It 6

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

Solution