Solve Using the Square Root Property (x-2)^2=45

${(x - 2)}^{2} = 45$

Step 1

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

$x - 2 = \pm \sqrt{45}$

Step 2

Simplify $\pm \sqrt{45}$ .

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$x - 2 = \pm \sqrt{9 (5)}$

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

$x - 2 = \pm \sqrt{3^{2} \cdot 5}$

Pull terms out from under the radical.

$x - 2 = \pm 3 \sqrt{5}$

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.

$x - 2 = 3 \sqrt{5}$

Add $2$ to both sides of the equation.

$x = 3 \sqrt{5} + 2$

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

$x - 2 = - 3 \sqrt{5}$

Add $2$ to both sides of the equation.

$x = - 3 \sqrt{5} + 2$

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

$x = 3 \sqrt{5} + 2, - 3 \sqrt{5} + 2$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 3 \sqrt{5} + 2, - 3 \sqrt{5} + 2$

Decimal Form:

$x = 8.70820393 \dots, - 4.70820393 \dots$