Find All Complex Solutions -3x^2-4x-5=0

$- 3 x^{2} - 4 x - 5 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = - 3$ , $b = - 4$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (- 3 \cdot - 5)}}{2 \cdot - 3}$

Step 3

Simplify.

Simplify the numerator.

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 3 \cdot - 5}}{2 \cdot - 3}$

Multiply $- 4 \cdot - 3 \cdot - 5$ .

Multiply $- 4$ by $- 3$ .

$x = \frac{4 \pm \sqrt{16 + 12 \cdot - 5}}{2 \cdot - 3}$

Multiply $12$ by $- 5$ .

$x = \frac{4 \pm \sqrt{16 - 60}}{2 \cdot - 3}$

Subtract $60$ from $16$ .

$x = \frac{4 \pm \sqrt{- 44}}{2 \cdot - 3}$

Rewrite $- 44$ as $- 1 (44)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 44}}{2 \cdot - 3}$

Rewrite $\sqrt{- 1 (44)}$ as $\sqrt{- 1} \cdot \sqrt{44}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{44}}{2 \cdot - 3}$

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

$x = \frac{4 \pm i \cdot \sqrt{44}}{2 \cdot - 3}$

Rewrite $44$ as $2^{2} \cdot 11$ .

Factor $4$ out of $44$ .

$x = \frac{4 \pm i \cdot \sqrt{4 (11)}}{2 \cdot - 3}$

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

$x = \frac{4 \pm i \cdot \sqrt{2^{2} \cdot 11}}{2 \cdot - 3}$

Pull terms out from under the radical.

$x = \frac{4 \pm i \cdot (2 \sqrt{11})}{2 \cdot - 3}$

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

$x = \frac{4 \pm 2 i \sqrt{11}}{2 \cdot - 3}$

Multiply $2$ by $- 3$ .

$x = \frac{4 \pm 2 i \sqrt{11}}{- 6}$

Simplify $\frac{4 \pm 2 i \sqrt{11}}{- 6}$ .

$x = \frac{2 \pm i \sqrt{11}}{- 3}$

Move the negative in front of the fraction.

$x = - \frac{2 \pm i \sqrt{11}}{3}$