# How do you solve 3z^2+10z+15=0?

Jun 15, 2015

The discriminant is negative, so I would use the quadratic formula directly to get:

$z = \frac{- 5 \pm 2 \sqrt{5} i}{3}$

#### Explanation:

$3 {z}^{2} + 10 z + 15$ is of the form $a {z}^{2} + b z + c$ with $a = 3$, $b = 10$ and $c = 15$.

This has discriminant $\Delta$ given by the formula:

$\Delta = {b}^{2} - 4 a c = {10}^{2} - \left(4 \times 3 \times 15\right) = 100 - 180 = - 80$

Since this is negative the quadratic equation has two distinct complex roots.

The roots are given by the quadratic formula:

$z = \frac{- b \pm \sqrt{\Delta}}{2 a} = \frac{- 10 \pm \sqrt{- 80}}{2 \cdot 3}$

$= \frac{- 10 \pm \sqrt{80} i}{6}$

$= \frac{- 10 \pm \sqrt{16 \cdot 5} i}{6}$

$= \frac{- 10 \pm 4 \sqrt{5} i}{6}$

$= \frac{- 5 \pm 2 \sqrt{5} i}{3}$