Solve for s s^2+5s=-20

$s^{2} + 5 s = - 20$

Step 1

Add $20$ to both sides of the equation.

$s^{2} + 5 s + 20 = 0$

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 $a = 1$ , $b = 5$ , and $c = 20$ into the quadratic formula and solve for $s$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (1 \cdot 20)}}{2 \cdot 1}$

Step 4

Simplify.

Simplify the numerator.

Raise $5$ to the power of $2$ .

$s = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot 20}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot 20$ .

Multiply $- 4$ by $1$ .

$s = \frac{- 5 \pm \sqrt{25 - 4 \cdot 20}}{2 \cdot 1}$

Multiply $- 4$ by $20$ .

$s = \frac{- 5 \pm \sqrt{25 - 80}}{2 \cdot 1}$

Subtract $80$ from $25$ .

$s = \frac{- 5 \pm \sqrt{- 55}}{2 \cdot 1}$

Rewrite $- 55$ as $- 1 (55)$ .

$s = \frac{- 5 \pm \sqrt{- 1 \cdot 55}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (55)}$ as $\sqrt{- 1} \cdot \sqrt{55}$ .

$s = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{55}}{2 \cdot 1}$

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

$s = \frac{- 5 \pm i \sqrt{55}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$s = \frac{- 5 \pm i \sqrt{55}}{2}$

Step 5

The final answer is the combination of both solutions.

$s = - \frac{5 - i \sqrt{55}}{2}, - \frac{5 + i \sqrt{55}}{2}$