Solve for z z^2=-9

$z^{2} = - 9$

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

$z = \pm \sqrt{- 9}$

Simplify $\pm \sqrt{- 9}$ .

Rewrite $- 9$ as $- 1 (9)$ .

$z = \pm \sqrt{- 1 (9)}$

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$z = \pm \sqrt{- 1} \cdot \sqrt{9}$

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

$z = \pm i \cdot \sqrt{9}$

Rewrite $9$ as $3^{2}$ .

$z = \pm i \cdot \sqrt{3^{2}}$

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

$z = \pm i \cdot 3$

Move $3$ to the left of $i$ .

$z = \pm 3 i$

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.

$z = 3 i$

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

$z = - 3 i$

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

$z = 3 i, - 3 i$