# How do you factor a^2 + b^2?

May 15, 2015

Whereas ${a}^{2} - {b}^{2} = \left(a + b\right) \left(a - b\right)$ is very simple, to factor ${a}^{2} + {b}^{2}$ requires the use of complex numbers.

If $i = \sqrt{- 1}$ then

$\left(a + i b\right) \left(a - i b\right)$

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

$= a - {i}^{2} b$

$= {a}^{2} - \left(- 1\right) {b}^{2}$

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

So ${a}^{2} + {b}^{2} = \left(a + i b\right) \left(a - i b\right)$, but there is no other factoring with real number coefficients.

May 13, 2018

${a}^{2} + {b}^{2}$ doesn't have a nice factorization over the reals, but over the complex numbers it's the squared magnitude of $a + b i ,$ which gives the factorization

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