Let S = z C : z^2 + i z = 0 . Then _ z S |z - 3 |^2 is equal to _______.

Step-by-step solution

The given equation is z^2 + i z = 0 . Clearly, z = 0 is a solution. For z 0 , multiply the equation by z to get z^3 + i z z = 0 z^3 + i|z|^2 = 0 . Taking the modulus on both sides, we get |z|^3 = |-i| |z|^2 |z|^3 = |z|^2 . Since z 0 , we have |z| = 1 . Substituting |z| = 1 back into the equation yields z^3 + i = 0 z^3 = -i . The non-zero roots are the three cube roots of -i . Let these be z₁, z₂, z₃ . The sum of the roots of z^3 + i = 0 is 0 , which implies Re (z₁) + Re (z₂) + Re (z₃) = 0 . We need to find _ z S |z