Solve Quadratic Equations Using the Square Root Property

The Square Root Property

With the $x^{2}$ term isolated, the square root property states that:

if $x^{2}=k$, then $x=\pm \sqrt{k}$

where $k$ is a nonzero real number.

HOW TO

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

1. Isolate the $x^{2}$ 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: $x^{2}=8$.

Solution

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

$\begin{aligned}x^{2} &=8 \\x &=\pm \sqrt{8} \\&=\pm 2 \sqrt{2}\end{aligned}$

The solutions are $2 \sqrt{2},-2 \sqrt{2}$.

Example 7

Solving a Quadratic Equation Using the Square Root Property

Solve the quadratic equation: $4 x^{2}+1=7$.

Solution

First, isolate the $x^{2}$ term. Then take the square root of both sides.

$\begin{aligned}4 x^{2}+1 &=7 \\4 x^{2} &=6 \\x^{2} &=\dfrac{6}{4} \\x &=\pm \dfrac{\sqrt{6}}{2}\end{aligned}$

The solutions are $\dfrac{\sqrt{6}}{2}$, and $-\dfrac{\sqrt{6}}{2}$.

Try It #6

Solve the quadratic equation using the square root property: $3(x-4)^{2}=15$.