# Math Help - Complex Factorisation

1. ## Complex Factorisation

Hi everyone, I was wondering if anyone can show me how these factorisations were carried out?

$
z^2 + i = (z + \frac{1}{\sqrt{2}}(1-i))(z - \frac{1}{\sqrt{2}}(1-i))
$

$
z^2 - i = (z + \frac{1}{\sqrt{2}}(1+i))(z - \frac{1}{\sqrt{2}}(1+i))
$

Any help would be much appreciated!

edit: $i$ is the imaginary unit, as always.

2nd edit: For anyone interested they arose while I was factorising this:

$x^4 + 1$

in order to evalute this:

$
\int_{-\infty}^{+\infty} \frac{e^{isx}}{x^4+1} dx
$

using contour integration.
I have the solution, I just can't replicate those factorisations.

2. $z^2=-i$

$z=\sqrt{-i}=e^{i/2(-\pi/2+2k\pi)};\; k=0,1$

same dif with the other one except the argument of $i$ is $\pi/2$ as well as:

$z^4=-1$

$z=(-1)^{1/4}=e^{i/4(\pi+2k\pi)};\; k=0,1,2,3$