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

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

Step 1

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

$x - 2 = \pm \sqrt{- 6}$

Step 2

Simplify $\pm \sqrt{- 6}$ .

Rewrite $- 6$ as $- 1 (6)$ .

$x - 2 = \pm \sqrt{- 1 (6)}$

Rewrite $\sqrt{- 1 (6)}$ as $\sqrt{- 1} \cdot \sqrt{6}$ .

$x - 2 = \pm \sqrt{- 1} \cdot \sqrt{6}$

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

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

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.

$x - 2 = i \sqrt{6}$

Add $2$ to both sides of the equation.

$x = i \sqrt{6} + 2$

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

$x - 2 = - i \sqrt{6}$

Add $2$ to both sides of the equation.

$x = - i \sqrt{6} + 2$

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

$x = i \sqrt{6} + 2, - i \sqrt{6} + 2$