Solve Using the Square Root Property (x-9)^2=18

${(x - 9)}^{2} = 18$

Step 1

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

$x - 9 = \pm \sqrt{18}$

Step 2

Simplify $\pm \sqrt{18}$ .

Rewrite $18$ as $3^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$x - 9 = \pm \sqrt{9 (2)}$

Rewrite $9$ as $3^{2}$ .

$x - 9 = \pm \sqrt{3^{2} \cdot 2}$

Pull terms out from under the radical.

$x - 9 = \pm 3 \sqrt{2}$

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.

$x - 9 = 3 \sqrt{2}$

Add $9$ to both sides of the equation.

$x = 3 \sqrt{2} + 9$

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

$x - 9 = - 3 \sqrt{2}$

Add $9$ to both sides of the equation.

$x = - 3 \sqrt{2} + 9$

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

$x = 3 \sqrt{2} + 9, - 3 \sqrt{2} + 9$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 3 \sqrt{2} + 9, - 3 \sqrt{2} + 9$

Decimal Form:

$x = 13.24264068 \dots, 4.75735931 \dots$