finding all the roots (including complex)of the equation

Question

Find all the roots of $z^4=16(z+2i)^4$.

Can someone help me teach/ guide to solve this equation?

There are lots of 4-th powers everywhere. Try taking 4-th roots of both sides. But remember what you have to do when taking a square root of a equation? That you get a $\pm 1$ term (in other words, two potential answers)? What should happen by taking 4-th roots? — zibadawa timmy, Nov 04 '13 at 07:05

score 1 · Answer 1 · answered Nov 04 '13 at 07:00

1

Hint : $z=2u(z+2i)$ , where $u=\{\pm1,\pm i\}$

answered Nov 04 '13 at 07:00

Lucian

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The OP did not ask for answers, but some guidance – Vishal Gupta Nov 04 '13 at 07:00
Too explicit, or too vague ? – Lucian Nov 04 '13 at 07:04
The OP could have got the answer from WolframAlpha. I think you need to explain step by step how you reached these answeres. I have not solved it myself though. – Vishal Gupta Nov 04 '13 at 07:05
There's nothing to ‘solve’, that's the whole point; all one has to do is to directly extract the fourth-root, with the mention that the four roots of unity also include $-1$, as well as $\pm i$. – Lucian Nov 04 '13 at 07:08
Yeah. Precisely. I did not even gaze at the question long enough. – Vishal Gupta Nov 04 '13 at 07:10
I think @aaa answer does precisely what OP wants. – Vishal Gupta Nov 04 '13 at 07:12

score 1 · Accepted Answer · answered Nov 04 '13 at 07:11

Just take the square root of both sides. Then affix with +/- to get 2 cases for that equation. Then for each case, get the square root again. Then affix the resulting equation with +/- to get 4 cases. Finally, solve for z for each of the 4 equations.

finding all the roots (including complex)of the equation

2 Answers2