Solve Using the Square Root Property (x-6)^2=-25

${(x - 6)}^{2} = - 25$

Step 1

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

$x - 6 = \pm \sqrt{- 25}$

Step 2

Simplify $\pm \sqrt{- 25}$ .

Rewrite $- 25$ as $- 1 (25)$ .

$x - 6 = \pm \sqrt{- 1 (25)}$

Rewrite $\sqrt{- 1 (25)}$ as $\sqrt{- 1} \cdot \sqrt{25}$ .

$x - 6 = \pm \sqrt{- 1} \cdot \sqrt{25}$

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

$x - 6 = \pm i \cdot \sqrt{25}$

Rewrite $25$ as $5^{2}$ .

$x - 6 = \pm i \cdot \sqrt{5^{2}}$

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

$x - 6 = \pm i \cdot 5$

Move $5$ to the left of $i$ .

$x - 6 = \pm 5 i$

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.

$x - 6 = 5 i$

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

Add $6$ to both sides of the equation.

$x = 5 i + 6$

Reorder $5 i$ and $6$ .

$x = 6 + 5 i$

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

$x - 6 = - 5 i$

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

Add $6$ to both sides of the equation.

$x = - 5 i + 6$

Reorder $- 5 i$ and $6$ .

$x = 6 - 5 i$

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

$x = 6 + 5 i, 6 - 5 i$