Solving $z^2-2iz+1=0$ in complex numbers

Question

Solve: $z^2-2iz+1=0$

I did: $$(z-i)^2-(i)^2+1=0$$ $$(z-i)^2+2=0$$ $$((z-i)-\sqrt{2})((z-i)+\sqrt{2})$$ but that's wrong. Why?

Answer 1

If by rt(2) you mean the squareroot of $2$, so $\sqrt{2}$, the problem is that you have to "put the $2$ on the other side" so you nee to take root of $-2$.

Or, recall $X^2 - a^2 = (X-a)(X+a)$ and so $X^2 + a^2 = (X-ia)(X+ia)$.