# How do you solve the equation: x^2 + 24x + 90 = 0?

Jul 3, 2015

Check the discriminant, then use the quadratic formula to find:

$x = - 12 \pm 3 \sqrt{6}$.

That is:

$x = - 12 - 3 \sqrt{6}$ or $x = - 12 + 3 \sqrt{6}$

#### Explanation:

$f \left(x\right) = {x}^{2} + 24 x + 90$ is of the form

$a {x}^{2} + b x + c$ with $a = 1$, $b = 24$ and $c = 90$.

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

$\Delta = {b}^{2} - 4 a c = {24}^{2} - \left(4 \times 1 \times 90\right) = 576 - 360$

$= 216 = {6}^{2} \cdot 6$

Since $\Delta$ is positive, but not a perfect square, $f \left(x\right) = 0$ has two distinct irrational real solutions, given by the quadratic formula:

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

$= \frac{- b \pm \sqrt{\Delta}}{2 a}$

$= \frac{- 24 \pm \sqrt{216}}{2}$

$= \frac{- 24 \pm 6 \sqrt{6}}{2}$

$= - 12 \pm 3 \sqrt{6}$