Right, this comes in 2 parts, the 1st part is that this problem is part (b) of a question, in part (a) you are given z= 3+5i however i don't know if you can/need to use this fact for this part of the question (gotta love my teachers ). second part is the question itself:
Solve the equation z^2 + 2z*= -3, where z* denotes the complex conjugate of z.
thanks in advance for any help, really appreciate it.
Another way to do it:
To solve
z² + 2z* = -3
Let z = (x + iy); z* = (x - iy)
(x + iy)² + 2(x-iy) = -3
x² + 2xiy -y² + 2x - 2iy = -3
Equating Real and Imaginary parts:
Real: x² + 2x - y² = -3
Imaginary: 2xiy - 2iy = 0
From imaginary part:
2iy (x-1) = 0
So x = 1 or y =0
Substituting y = 0 into the real part:
x² + 2x + 3 = 0.
This quadratic has no real roots. We know that x has to be an integer, so this will not give us a solution.
Substituting x = 1 into the real part:
y² = 6
y= √6
Thus z = x + iy = 1 + √6i