Solve Using the Square Root Property (x-7)^2=32

${(x - 7)}^{2} = 32$

Step 1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x - 7 = \pm \sqrt{32}$

Step 2

Simplify $\pm \sqrt{32}$ .

Rewrite $32$ as $4^{2} \cdot 2$ .

Factor $16$ out of $32$ .

$x - 7 = \pm \sqrt{16 (2)}$

Rewrite $16$ as $4^{2}$ .

$x - 7 = \pm \sqrt{4^{2} \cdot 2}$

Pull terms out from under the radical.

$x - 7 = \pm 4 \sqrt{2}$

Step 3

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x - 7 = 4 \sqrt{2}$

Add $7$ to both sides of the equation.

$x = 4 \sqrt{2} + 7$

Next, use the negative value of the $\pm$ to find the second solution.

$x - 7 = - 4 \sqrt{2}$

Add $7$ to both sides of the equation.

$x = - 4 \sqrt{2} + 7$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 4 \sqrt{2} + 7, - 4 \sqrt{2} + 7$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 4 \sqrt{2} + 7, - 4 \sqrt{2} + 7$

Decimal Form:

$x = 12.65685424 \dots, 1.34314575 \dots$