Solve the equation: z^2 + |z| =0 where z=x+ iy.

Question

Solve the equation: z²+|z|=0 where z=x+iy (x,y ∈ R and i²=-1)

Answer 1

If z² + |z| = 0, then z² = -|z|. Taking the modulus on both sides, |z²| = |z|, or |z|² = |z|. Hence, |z| = 0 or 1. If |z| = 0, then z = 0. If |z| = 1, then z^2 = -1, resulting in z = ±i. So the solutions are 0, i, -i.