Solve for z (z+4/3)^2=-3

${(z + \frac{4}{3})}^{2} = - 3$

Step 1

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

$z + \frac{4}{3} = \pm \sqrt{- 3}$

Step 2

Simplify $\pm \sqrt{- 3}$ .

Rewrite $- 3$ as $- 1 (3)$ .

$z + \frac{4}{3} = \pm \sqrt{- 1 (3)}$

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$z + \frac{4}{3} = \pm \sqrt{- 1} \cdot \sqrt{3}$

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

$z + \frac{4}{3} = \pm i \sqrt{3}$

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.

$z + \frac{4}{3} = i \sqrt{3}$

Subtract $\frac{4}{3}$ from both sides of the equation.

$z = i \sqrt{3} - \frac{4}{3}$

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

$z + \frac{4}{3} = - i \sqrt{3}$

Subtract $\frac{4}{3}$ from both sides of the equation.

$z = - i \sqrt{3} - \frac{4}{3}$

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

$z = i \sqrt{3} - \frac{4}{3}, - i \sqrt{3} - \frac{4}{3}$