Solve for z 5z^2+32z=-155

$5 z^{2} + 32 z = - 155$

Step 1

Add $155$ to both sides of the equation.

$5 z^{2} + 32 z + 155 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 5$ , $b = 32$ , and $c = 155$ into the quadratic formula and solve for $z$ .

$\frac{- 32 \pm \sqrt{32^{2} - 4 \cdot (5 \cdot 155)}}{2 \cdot 5}$

Step 4

Simplify.

Simplify the numerator.

Raise $32$ to the power of $2$ .

$z = \frac{- 32 \pm \sqrt{1024 - 4 \cdot 5 \cdot 155}}{2 \cdot 5}$

Multiply $- 4 \cdot 5 \cdot 155$ .

Multiply $- 4$ by $5$ .

$z = \frac{- 32 \pm \sqrt{1024 - 20 \cdot 155}}{2 \cdot 5}$

Multiply $- 20$ by $155$ .

$z = \frac{- 32 \pm \sqrt{1024 - 3100}}{2 \cdot 5}$

Subtract $3100$ from $1024$ .

$z = \frac{- 32 \pm \sqrt{- 2076}}{2 \cdot 5}$

Rewrite $- 2076$ as $- 1 (2076)$ .

$z = \frac{- 32 \pm \sqrt{- 1 \cdot 2076}}{2 \cdot 5}$

Rewrite $\sqrt{- 1 (2076)}$ as $\sqrt{- 1} \cdot \sqrt{2076}$ .

$z = \frac{- 32 \pm \sqrt{- 1} \cdot \sqrt{2076}}{2 \cdot 5}$

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

$z = \frac{- 32 \pm i \cdot \sqrt{2076}}{2 \cdot 5}$

Rewrite $2076$ as $2^{2} \cdot 519$ .

Factor $4$ out of $2076$ .

$z = \frac{- 32 \pm i \cdot \sqrt{4 (519)}}{2 \cdot 5}$

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

$z = \frac{- 32 \pm i \cdot \sqrt{2^{2} \cdot 519}}{2 \cdot 5}$

Pull terms out from under the radical.

$z = \frac{- 32 \pm i \cdot (2 \sqrt{519})}{2 \cdot 5}$

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

$z = \frac{- 32 \pm 2 i \sqrt{519}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$z = \frac{- 32 \pm 2 i \sqrt{519}}{10}$

Simplify $\frac{- 32 \pm 2 i \sqrt{519}}{10}$ .

$z = \frac{- 16 \pm i \sqrt{519}}{5}$

Step 5

The final answer is the combination of both solutions.

$z = - \frac{16 - i \sqrt{519}}{5}, - \frac{16 + i \sqrt{519}}{5}$