Solve Using the Square Root Property (x-2)^2=-4

${(x - 2)}^{2} = - 4$

Step 1

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

$x - 2 = \pm \sqrt{- 4}$

Step 2

Simplify $\pm \sqrt{- 4}$ .

Rewrite $- 4$ as $- 1 (4)$ .

$x - 2 = \pm \sqrt{- 1 (4)}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x - 2 = \pm \sqrt{- 1} \cdot \sqrt{4}$

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

$x - 2 = \pm i \cdot \sqrt{4}$

Rewrite $4$ as $2^{2}$ .

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

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

$x - 2 = \pm i \cdot 2$

Move $2$ to the left of $i$ .

$x - 2 = \pm 2 i$

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.

$x - 2 = 2 i$

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

Add $2$ to both sides of the equation.

$x = 2 i + 2$

Reorder $2 i$ and $2$ .

$x = 2 + 2 i$

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

$x - 2 = - 2 i$

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

Add $2$ to both sides of the equation.

$x = - 2 i + 2$

Reorder $- 2 i$ and $2$ .

$x = 2 - 2 i$

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

$x = 2 + 2 i, 2 - 2 i$