Explanation of a factoring process

Question

I have found in a solved exercise the following:

$$x^{-2}-\sqrt{2}x^{-1}+1=(1-e^{j\frac{\pi}{4}}\cdot x^{-1})(1-e^{-j\frac{\pi}{4}}\cdot x^{-1})$$

Can someone explain how did the exponentials appear and how they end up being multiplied with $x^{-1}$?

Answer 1

Set $u=x^{-1}$ and solve the quadratic equation $u^2-\sqrt2u+1=0$: $\Delta=-2$, so the roots are: $$z_1,z_2=\frac{\sqrt 2\pm \mathrm i\sqrt2}2=\mathrm e^{\pm\tfrac{\mathrm i\pi}4},$$ whence the factorisation $$u^2-\sqrt2u+1=\Bigl(u-\mathrm e^{\tfrac{\mathrm i\pi}4}\Bigr)\Bigl(u-\mathrm e^{-\tfrac{\mathrm i\pi}4}\Bigr).$$ Then replace $u$ with $x^{-1}$.

Answer 2

This is coming from $$\frac{1}{\sqrt{2}}(1+j)=e^{j\frac{\pi}{4}}$$