1. The problem statement, all variables and given/known data Find real x and y, for which |z+3|=1-iz 2. Relevant equations z=x+iy=reiθ=rcos(θ) + rsin(θ)i 3. The attempt at a solution I know that there must exist some x and y which satisfies both of these equations, since the real part of the LHS must be the same as the RHS, and same with the imaginary part. What is tripping me up is that z is on both sides. If there wasn't a z on either side, I would normally just convert each side into it's polar form, and solve for r and θ, thus finding x and y. In this case, I've only gotten so far as changing to polar and doing trivial manipulations to both sides: reiθ +3 = 1-ireiθ reiθ (1+i) = -2 reiθ = i-1 = (i-1)ei2πn Then I found r=√2, θ=2π. So x=√2 and y =0. That's where I'm at so far. I ignored the absolute value of the LHS, which I'm sure probably plays a part in the answer, but I don't quite know how. The answer in the back of the book is x=0 and y=4. I have no idea how to get that answer. Any pointers would be greatly appreciated.