Solve Using the Square Root Property (r-7)^2=-2

${(r - 7)}^{2} = - 2$

Step 1

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

$r - 7 = \pm \sqrt{- 2}$

Step 2

Simplify $\pm \sqrt{- 2}$ .

Rewrite $- 2$ as $- 1 (2)$ .

$r - 7 = \pm \sqrt{- 1 (2)}$

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$r - 7 = \pm \sqrt{- 1} \cdot \sqrt{2}$

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

$r - 7 = \pm i \sqrt{2}$

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.

$r - 7 = i \sqrt{2}$

Add $7$ to both sides of the equation.

$r = i \sqrt{2} + 7$

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

$r - 7 = - i \sqrt{2}$

Add $7$ to both sides of the equation.

$r = - i \sqrt{2} + 7$

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

$r = i \sqrt{2} + 7, - i \sqrt{2} + 7$