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?

  • 2
    $\begingroup$ You have $4$ almost mentioned, $\pm\sqrt{3\pm 4i}$. Probably you should give the answers in exponential form. $\endgroup$ – André Nicolas Mar 22 '14 at 1:30
  • $\begingroup$ Continue simplifying your results. Working your last expression with the $\pm$ as suggested by André Nicolas, you should end with roots : $2+i,2-i,-2+i,-2-i$. $\endgroup$ – Claude Leibovici Mar 22 '14 at 5:01
  • $\begingroup$ Thank @ClaudeLeibovici for your comment. I think that I am supposed to have these result: $\sqrt{3+4i}$, $\sqrt{3-4i}$, $-\sqrt{3+4i}$, $-\sqrt{3-4i}$. How you came up with your roots? $\endgroup$ – bman Mar 22 '14 at 6:10
  • $\begingroup$ Just computing the square root of a complex number ! Use the trigonometric representation. $\endgroup$ – Claude Leibovici Mar 22 '14 at 6:13

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.

  • $\begingroup$ Wow, I feel so dumb to forget to put pm sign when I squared root the number. Thank you. $\endgroup$ – bman Mar 22 '14 at 3:10

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