# How do you factor 45r^2-120rs+80s^2?

Mar 30, 2016

$45 {r}^{2} - 120 r s + 80 {s}^{2} = 5 {\left(3 r - 4 s\right)}^{2}$

#### Explanation:

First separate out the common scalar factor $5$:

$45 {r}^{2} - 120 r s + 80 {s}^{2} = 5 \left(9 {r}^{2} - 24 r s + 16 {s}^{2}\right)$

Next note that both $9 {r}^{2} = {\left(3 r\right)}^{2}$ and $16 {s}^{2} = {\left(4 s\right)}^{2}$ are perfect squares. So given the middle minus sign check to see if this is the square of $\left(3 r - 4 s\right)$:

${\left(3 r - 4 s\right)}^{2} = {\left(3 r\right)}^{2} - 2 \left(3 r\right) \left(4 s\right) + {\left(4 s\right)}^{2} = 9 {r}^{2} - 24 r s + 16 {s}^{2}$

So $9 {r}^{2} - 24 r s + 16 {s}^{2}$ is a perfect square trinomial.