# How do you factor 5c^2-24cd-5d^2?

May 9, 2016

$5 {c}^{2} - 24 c d - 5 {d}^{2} = \left(c - 5 d\right) \left(5 c + d\right)$

#### Explanation:

Here's one way...

Multiply through by $5$, complete the square, use the difference of squares identity:

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

with $a = \left(5 c - 12 d\right)$ and $b = 13 d$, then divide by $5$...

$5 \left(5 {c}^{2} - 24 c d - 5 {d}^{2}\right)$

$= 25 {c}^{2} - 120 c d - 25 {d}^{2}$

$= {\left(5 c - 12 d\right)}^{2} - {\left(12 d\right)}^{2} - 25 {d}^{2}$

$= {\left(5 c - 12 d\right)}^{2} - \left(144 + 25\right) {d}^{2}$

$= {\left(5 c - 12 d\right)}^{2} - 169 {d}^{2}$

$= {\left(5 c - 12 d\right)}^{2} - {\left(13 d\right)}^{2}$

$= \left(\left(5 c - 12 d\right) - 13 d\right) \left(\left(5 c - 12 d\right) + 13 d\right)$

$= \left(5 c - 25 d\right) \left(5 c + d\right)$

$= 5 \left(c - 5 d\right) \left(5 c + d\right)$

So:

$5 {c}^{2} - 24 c d - 5 {d}^{2} = \left(c - 5 d\right) \left(5 c + d\right)$