# How do you factor 60x^2 - 124xy + 63y^2?

May 28, 2015

Let's try a version of the AC Method:

$A = 60$, $B = 124$, $C = 63$

We are looking for a pair of factors of $A C = 60 \times 63$ which add up to $B = 124$.

$A C = 60 \cdot 63 = {2}^{2} \cdot {3}^{3} \cdot 5 \cdot 7$ has quite a few possible pairs of factors.

We can narrow down the search a bit: Since the sum is even, the factors of $2$ must be split between them.

So look for a pair of factors of ${3}^{3} \cdot 5 \cdot 7$ that add to $62$.
Well ${3}^{3} = 27$ and $5 \cdot 7 = 35$ work.

So the original pair we were looking for is $2 \cdot 27 = 54$ and $2 \cdot 35 = 70$

Use this pair to split the middle term, then factor by grouping:

$60 {x}^{2} - 124 x y + 63 {y}^{2}$

$= 60 {x}^{2} - 54 x y - 70 x y + 63 {y}^{2}$

$= \left(60 {x}^{2} - 54 x y\right) - \left(70 x y - 63 {y}^{2}\right)$

$= 6 x \left(10 x - 9 y\right) - 7 y \left(10 x - 9 y\right)$

$= \left(6 x - 7 y\right) \left(10 x - 9 y\right)$