Solve Using the Square Root Property (x-9)^2=24

${(x - 9)}^{2} = 24$

Step 1

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

$x - 9 = \pm \sqrt{24}$

Step 2

Simplify $\pm \sqrt{24}$ .

Rewrite $24$ as $2^{2} \cdot 6$ .

Factor $4$ out of $24$ .

$x - 9 = \pm \sqrt{4 (6)}$

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

$x - 9 = \pm \sqrt{2^{2} \cdot 6}$

Pull terms out from under the radical.

$x - 9 = \pm 2 \sqrt{6}$

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.

$x - 9 = 2 \sqrt{6}$

Add $9$ to both sides of the equation.

$x = 2 \sqrt{6} + 9$

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

$x - 9 = - 2 \sqrt{6}$

Add $9$ to both sides of the equation.

$x = - 2 \sqrt{6} + 9$

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

$x = 2 \sqrt{6} + 9, - 2 \sqrt{6} + 9$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 2 \sqrt{6} + 9, - 2 \sqrt{6} + 9$

Decimal Form:

$x = 13.89897948 \dots, 4.10102051 \dots$