Solve Using the Square Root Property (x-22)^2=12

${(x - 22)}^{2} = 12$

Step 1

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

$x - 22 = \pm \sqrt{12}$

Step 2

Simplify $\pm \sqrt{12}$ .

Rewrite $12$ as $2^{2} \cdot 3$ .

Factor $4$ out of $12$ .

$x - 22 = \pm \sqrt{4 (3)}$

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

$x - 22 = \pm \sqrt{2^{2} \cdot 3}$

Pull terms out from under the radical.

$x - 22 = \pm 2 \sqrt{3}$

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.

$x - 22 = 2 \sqrt{3}$

Add $22$ to both sides of the equation.

$x = 2 \sqrt{3} + 22$

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

$x - 22 = - 2 \sqrt{3}$

Add $22$ to both sides of the equation.

$x = - 2 \sqrt{3} + 22$

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

$x = 2 \sqrt{3} + 22, - 2 \sqrt{3} + 22$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 2 \sqrt{3} + 22, - 2 \sqrt{3} + 22$

Decimal Form:

$x = 25.46410161 \dots, 18.53589838 \dots$