Solve Using the Square Root Property (x-9)^2=-5

${(x - 9)}^{2} = - 5$

Step 1

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

$x - 9 = \pm \sqrt{- 5}$

Step 2

Simplify $\pm \sqrt{- 5}$ .

Rewrite $- 5$ as $- 1 (5)$ .

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

Rewrite $\sqrt{- 1 (5)}$ as $\sqrt{- 1} \cdot \sqrt{5}$ .

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

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

$x - 9 = \pm i \sqrt{5}$

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.

$x - 9 = i \sqrt{5}$

Add $9$ to both sides of the equation.

$x = i \sqrt{5} + 9$

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

$x - 9 = - i \sqrt{5}$

Add $9$ to both sides of the equation.

$x = - i \sqrt{5} + 9$

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

$x = i \sqrt{5} + 9, - i \sqrt{5} + 9$