Solve Using the Square Root Property (y+2)^2+152=0

${(y + 2)}^{2} + 152 = 0$

Step 1

Subtract $152$ from both sides of the equation.

${(y + 2)}^{2} = - 152$

Step 2

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

$y + 2 = \pm \sqrt{- 152}$

Step 3

Simplify $\pm \sqrt{- 152}$ .

Rewrite $- 152$ as $- 1 (152)$ .

$y + 2 = \pm \sqrt{- 1 (152)}$

Rewrite $\sqrt{- 1 (152)}$ as $\sqrt{- 1} \cdot \sqrt{152}$ .

$y + 2 = \pm \sqrt{- 1} \cdot \sqrt{152}$

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

$y + 2 = \pm i \cdot \sqrt{152}$

Rewrite $152$ as $2^{2} \cdot 38$ .

Factor $4$ out of $152$ .

$y + 2 = \pm i \cdot \sqrt{4 (38)}$

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

$y + 2 = \pm i \cdot \sqrt{2^{2} \cdot 38}$

Pull terms out from under the radical.

$y + 2 = \pm i \cdot (2 \sqrt{38})$

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

$y + 2 = \pm 2 i \sqrt{38}$

Step 4

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.

$y + 2 = 2 i \sqrt{38}$

Subtract $2$ from both sides of the equation.

$y = 2 i \sqrt{38} - 2$

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

$y + 2 = - 2 i \sqrt{38}$

Subtract $2$ from both sides of the equation.

$y = - 2 i \sqrt{38} - 2$

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

$y = 2 i \sqrt{38} - 2, - 2 i \sqrt{38} - 2$