Solve Using the Square Root Property (x-8)^2=-64

${(x - 8)}^{2} = - 64$

Step 1

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

$x - 8 = \pm \sqrt{- 64}$

Step 2

Simplify $\pm \sqrt{- 64}$ .

Rewrite $- 64$ as $- 1 (64)$ .

$x - 8 = \pm \sqrt{- 1 (64)}$

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$x - 8 = \pm \sqrt{- 1} \cdot \sqrt{64}$

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

$x - 8 = \pm i \cdot \sqrt{64}$

Rewrite $64$ as $8^{2}$ .

$x - 8 = \pm i \cdot \sqrt{8^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x - 8 = \pm i \cdot 8$

Move $8$ to the left of $i$ .

$x - 8 = \pm 8 i$

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.

$x - 8 = 8 i$

Move all terms not containing $x$ to the right side of the equation.

Add $8$ to both sides of the equation.

$x = 8 i + 8$

Reorder $8 i$ and $8$ .

$x = 8 + 8 i$

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

$x - 8 = - 8 i$

Move all terms not containing $x$ to the right side of the equation.

Add $8$ to both sides of the equation.

$x = - 8 i + 8$

Reorder $- 8 i$ and $8$ .

$x = 8 - 8 i$

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

$x = 8 + 8 i, 8 - 8 i$