SOLUTION: Find all complex numbers z such that |z|^2-2\bar{z}+iz=2i

Question 1085367: Find all complex numbers z such that
|z|^2-2\bar{z}+iz=2i
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
I'm assuming the initial equation is

If so, then let where

(a,b are real numbers)

The complex conjugate of is

Using the pythagorean theorem, and a visual representation of , we can say

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Using those equations, we can make a bit of substitutions and rearrangements to get the following

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From that last equation, in the section above, we can equate the real and imaginary parts to form these two equations

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Solve for 'a'

Now plug this into and solve for b

or

or

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If , then

Therefore one solution is which is purely a real number.

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If , then

The other solution is which is a purely imaginary number

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Summary:

The two solutions are (which is the same as ) and (which is the same as )
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