3

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?

bman
  • 675
  • 2
    You have $4$ almost mentioned, $\pm\sqrt{3\pm 4i}$. Probably you should give the answers in exponential form. – André Nicolas Mar 22 '14 at 01:30
  • Continue simplifying your results. Working your last expression with the $\pm$ as suggested by André Nicolas, you should end with roots : $2+i,2-i,-2+i,-2-i$. – Claude Leibovici Mar 22 '14 at 05:01
  • 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? – bman Mar 22 '14 at 06:10
  • Just computing the square root of a complex number ! Use the trigonometric representation. – Claude Leibovici Mar 22 '14 at 06:13

1 Answers1

4

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.

TheBluegrassMathematician
  • 826
  • Wow, I feel so dumb to forget to put pm sign when I squared root the number. Thank you. – bman Mar 22 '14 at 03:10