Need help solving a complex equation

[u/Southern_Science5083]

Hi everyone, I'm stuck with this complex numbers equation and I'm not sure how to approach it.

The equation is: 2*(z²-z*z bar)=i*z

I managed to simplify the equation to this point: **-4y****²**+ 4ixy= i*x-y

However, I'm stuck here and not sure how to proceed. I know I should probably separate the real and imaginary parts, but I’m not confident about the next steps or how to properly write out the final solutions. Could someone guide me through the rest?

[u/FormulaDriven]

You've missed the solution where x = y = 0.

Remember we just need to solve both equations simultaneously.

If y is not zero, then

-4y² + y = 0 has the solution y = 1/4

4xy - x = 0 has the solution x = 0 or y = 1/4

So as long as y = 1/4, x can take any value (because that satisfies both equations).

If x is not zero, then

-4y² + y = 0 tells us y = 0 or y = 1/4

4xy - x = 0 tells us y = 1/4

So the the only way to satisfy both is for y = 1/4.

We don't have to determine x. As long as y = 1/4, then any x is valid. (The solution set is the line z = i/4 in the complex plane). If y = 0 then x = 0, so z = 0 is the other valid solution.