# How do you factor 18+8r^2-30r?

May 25, 2015

$18 + 8 {r}^{2} - 30 r$

$= 2 \left(4 {r}^{2} - 15 r + 9\right)$

$4 {r}^{2} - 15 r + 9$ is in the form $a {r}^{2} + b r + c$, with $a = 4$, $b = - 15$ and $c = 9$.

This has discriminant given by the formula:

$\Delta = {b}^{2} - 4 a c = {\left(- 15\right)}^{2} - \left(4 \times 4 \times 9\right)$

$= 225 - 144 = 81 = {9}^{2}$

Since this is a perfect square, the quadratic equation $4 {r}^{2} - 15 r + 9 = 0$ has two distinct real rational roots, given by the formula:

$r = \frac{- b \pm \sqrt{\Delta}}{2 a} = \frac{15 \pm 9}{8}$

That is $r = \frac{3}{4}$ and $r = 3$

From this we can deduce:

$4 {r}^{2} - 15 r + 9 = \left(4 r - 3\right) \left(r - 3\right)$

So:

$18 + 8 {r}^{2} - 30 r = 2 \left(4 r - 3\right) \left(r - 3\right)$