How do you factor a^2-b^2+6b-9?

May 19, 2017

${a}^{2} - {b}^{2} + 6 b - 9 = \left(a - b + 3\right) \left(a + b - 3\right)$

Explanation:

The difference of squares identity can be written:

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

We can use this with $A = a$ and $B = b - 3$ as follows:

${a}^{2} - {b}^{2} + 6 b - 9 = {a}^{2} - \left({b}^{2} - 6 b + 9\right)$

$\textcolor{w h i t e}{{a}^{2} - {b}^{2} + 6 b - 9} = {a}^{2} - {\left(b - 3\right)}^{2}$

$\textcolor{w h i t e}{{a}^{2} - {b}^{2} + 6 b - 9} = \left(a - \left(b - 3\right)\right) \left(a + \left(b - 3\right)\right)$

$\textcolor{w h i t e}{{a}^{2} - {b}^{2} + 6 b - 9} = \left(a - b + 3\right) \left(a + b - 3\right)$