Solve Using the Square Root Property (x-5)^2=-2

${(x - 5)}^{2} = - 2$

Step 1

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

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

Step 2

Simplify $\pm \sqrt{- 2}$ .

Rewrite $- 2$ as $- 1 (2)$ .

$x - 5 = \pm \sqrt{- 1 (2)}$

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

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

Rewrite $\sqrt{- 1}$ as $i$ .

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

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.

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

Add $5$ to both sides of the equation.

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

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

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

Add $5$ to both sides of the equation.

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

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

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