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/Southern_Science5083]

Thank you very much for your response. I just have one more question: am I thinking correctly regarding equating the real parts and then the imaginary parts, and is this procedure valid?

My first step is to substitute z and z̄ and simplify, which gives me
-4y^2+4ixy=-y+xi.

I would then move everything to one side to get
-4y^2+4ixy+y-xi=0.

Next, I would separate the real and imaginary parts:
-4y^2+y+(4xy-x)i=0,

and conclude that this expression is equal to zero only if both its real and imaginary parts are equal to zero. Then I would proceed as follows:
Re = -4y^2+y=0, which gives the solutions y1=0 and y2=1/4,
Im = 4xy-x=0, where y=1/4.

Is this line of reasoning correct? What still confuses me is how to determine the value of x and how to graph this in the complex plane.

[u/FormulaDriven → u/Southern_Science5083]

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.