How do you factor 12r^2-r-63?

Jul 30, 2016

$\left(3 r - 7\right) \left(4 r + 9\right)$

Explanation:

While this might appear daunting at first, there is an easy way to get to the correct factors.

We are looking for factors of 12 and 63 which combine to give a difference of 1.

$12 \times 63 = 756$, which has a huge number of factors!

However, as the difference in the factors is only 1, the factors are very close to $\sqrt{756}$

$\sqrt{756} = 27.495 \ldots . .$

Use factors of 12 and 63 to find factors which make 27 and 28
There must be more negatives.
Use the cross products

$\text{ 3 7 } \Rightarrow 4 \times 7 = 28$
$\text{ 4 9 "rArr 3 xx 9 = 27" } 28 - 27 = 1$

$\left(3 r - 7\right) \left(4 r + 9\right)$