Simple complex numbers question

[gtg177i]

1. The problem statement, all variables and given/known data
z+(conjugate of z)^2=4


2. Relevant equations
z = x+iy
(x+iy)+(x-iy)^2=4


3. The attempt at a solution
the solutions give (x+iy)+(x-iy)^2= x + x^2 - y^2. how do they reach that?
I get (x+iy)+(x-iy)^2 = x + iy + x^2 -2xiy + i^2*y^2.
I think the question is the ( conjugate of z )^2. e.g. z with the line on top of it and that squared.
Thanks!

[Mark44]

1. The problem statement, all variables and given/known data
z+(conjugate of z)^2=4


2. Relevant equations
z = x+iy
(x+iy)+(x-iy)^2=4


3. The attempt at a solution
the solutions give (x+iy)+(x-iy)^2= x + x^2 - y^2. how do they reach that?

They aren't showing all their steps.
x + iy + (x - iy)^2 = 4
==> x + iy + x^2 - y^2 -i2xy = 4
==> x + x^2 - y^2 + i(y - 2xy) = 4
Since the imaginary part of 4 is 0, it must be that y - 2xy = 0, or y(1 - 2x) = 0, which happens if y = 0 or if x = 1/2.

Then x + x^2 - y^2 = 4 and (y = 0 or x = 1/2)

I get (x+iy)+(x-iy)^2 = x + iy + x^2 -2xiy + i^2*y^2.
I think the question is the ( conjugate of z )^2. e.g. z with the line on top of it and that squared.
Thanks!