Solve Using the Square Root Property (x-1)^2=45

${(x - 1)}^{2} = 45$

Step 1

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

$x - 1 = \pm \sqrt{45}$

Step 2

Simplify $\pm \sqrt{45}$ .

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

Factor $9$ out of $45$ .

$x - 1 = \pm \sqrt{9 (5)}$

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

$x - 1 = \pm \sqrt{3^{2} \cdot 5}$

Pull terms out from under the radical.

$x - 1 = \pm 3 \sqrt{5}$

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.

$x - 1 = 3 \sqrt{5}$

Add $1$ to both sides of the equation.

$x = 3 \sqrt{5} + 1$

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

$x - 1 = - 3 \sqrt{5}$

Add $1$ to both sides of the equation.

$x = - 3 \sqrt{5} + 1$

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

$x = 3 \sqrt{5} + 1, - 3 \sqrt{5} + 1$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 3 \sqrt{5} + 1, - 3 \sqrt{5} + 1$

Decimal Form:

$x = 7.70820393 \dots, - 5.70820393 \dots$