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

Mar 28, 2016

Complete the square to find:

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

#### Explanation:

This can be solved by completing the square.

Also use the difference of squares identity:

${a}^{2} - {b}^{2} = \left(a - b\right) \left(a + b\right)$

with $a = \left(x + 12\right)$ and $b = 3 \sqrt{6}$ as follows:

$0 = {x}^{2} + 24 x + 90$

$= {\left(x + 12\right)}^{2} - 144 + 90$

$= {\left(x + 12\right)}^{2} - 54$

$= {\left(x + 12\right)}^{2} - \left({3}^{2} \cdot 6\right)$

$= {\left(x + 12\right)}^{2} - {\left(3 \sqrt{6}\right)}^{2}$

$= \left(\left(x + 12\right) - 3 \sqrt{6}\right) \left(\left(x + 12\right) + 3 \sqrt{6}\right)$

$= \left(x + 12 - 3 \sqrt{6}\right) \left(x + 12 + 3 \sqrt{6}\right)$

Hence:

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