complex numbers - How to solve quadratic function with degree higher than two?

I am struggling to solve the function $z^4 - 6z^2 + 25 = 0$ mostly because it has a degree of $4$. This is my solution so far:

Let $y = z^2 \Longrightarrow y^2 - 6y + 25 = 0$.

Now when we solve for y we get: $y=3 \pm 4i$.

So $z^2 = 3 \pm 4i$. Consequently $z = \sqrt{3 \pm 4i}$

But I know this is not the right answer, because a quadratic equation with degree four is supposed to have four answers. But I only get one answer. What I am doing wrong?

Answer

You are very close to the answer. Just put a plus or minus in front of the solution and you have your complete answer. It's always the simple things.