# How do you factor completely x^2+2ax = b^2-a^2?

Mar 21, 2018

In factored form this is:

$\left(x + a - b\right) \left(x + a + b\right) = 0$

#### Explanation:

The difference of squares identity can be written:

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

We will use this with $A = \left(x + a\right)$ and $B = b$.

Given:

${x}^{2} + 2 a x = {b}^{2} - {a}^{2}$

Add ${a}^{2} - {b}^{2}$ to both sides to get:

$0 = {x}^{2} + 2 a x + {a}^{2} - {b}^{2}$

$\textcolor{w h i t e}{0} = {\left(x + a\right)}^{2} - {b}^{2}$

$\textcolor{w h i t e}{0} = \left(\left(x + a\right) - b\right) \left(\left(x + a\right) + b\right)$

$\textcolor{w h i t e}{0} = \left(x + a - b\right) \left(x + a + b\right)$