solve $x^2+2ix+1=0$

Question

Hey folks I'm trying to solve $x^2+2ix+1=0$.

Squaring it ($x^2+2ix=-1)^2 \implies x^4-4x=1 \implies$ ... leads nowhere

Factoring it ($x^2+2ix+1=0)^2 \implies x(x+i)+(x+1)=0 \implies$ ... leads nowhere

I know that the answer is $(-1\pm \sqrt2)i$. Any hints are appreciated.

Answer 1

Completing the square is the fastest here: $\quad x^2+2ix+1=(x+i)^2+2 $, so $$x^2+2ix+1=0\iff (x+i)^2=-2 \iff x+i=\pm\sqrt 2 i\iff x=(-1\pm\sqrt 2)i.$$

Answer 2

Hint

$$x^2+2ix+(i)^2=-2 \\ (x+i)^2=-2$$