# How do you factor a^2+4ab-3b^2 ?

Jun 28, 2015

Complete the square to find:

${a}^{2} + 4 a b - 3 {b}^{2} = \left(a + \left(2 - \sqrt{7}\right) b\right) \left(a + \left(2 + \sqrt{7}\right) b\right)$

#### Explanation:

${\left(a + 2 b\right)}^{2} = {a}^{2} + 4 a b + 4 {b}^{2}$

So

${a}^{2} + 4 a b - 3 {b}^{2}$

$= {\left(a + 2 b\right)}^{2} - 4 {b}^{2} - 3 {b}^{2}$

$= {\left(a + 2 b\right)}^{2} - 7 {b}^{2}$

$= {\left(a + 2 b\right)}^{2} - {\left(\sqrt{7} b\right)}^{2}$

$= \left(\left(a + 2 b\right) - \sqrt{7} b\right) \left(\left(a + 2 b\right) + \sqrt{7} b\right)$

$= \left(a + \left(2 - \sqrt{7}\right) b\right) \left(a + \left(2 + \sqrt{7}\right) b\right)$

...using the difference of squares identity

${A}^{2} - {B}^{2} = \left(A - B\right) \left(A + B\right)$

with $A = a + 2 b$ and $B = \sqrt{7} b$