# How do you factor quadratic equations with two variables?

Mar 28, 2015

$4 {x}^{2} - 9 {y}^{2} = \left(2 x + 3 y\right) \left(2 x - 3 y\right)$ is a special product.
As is
$25 {x}^{2} - 30 x y + 9 {y}^{2} = {\left(5 x - 3 y\right)}^{2}$

$12 {x}^{2} - 9 x y + 6 y$ has a common factor of $3$, but that's all.

${x}^{2} - 4 x y - 5 {y}^{2}$ is factored by trial and error.

If it is easily factorable, it must look like $\left(x + a y\right) \left(x + b y\right)$

where $a b = 5$ and $a + b = - 4$

It's really a lot like factoring: ${x}^{2} - 4 x - 5$

${x}^{2} - 4 x - 5 = \left(x - 5\right) \left(x + 1\right)$

${x}^{2} - 4 x y - 5 {y}^{2} = \left(x - 5 y\right) \left(x + y\right)$