Question

Let z be a complex number such that the imaginary part of z is nonzero and A = z^2 + z + 1 is real. Then A cannot take the value of what?

Answer 1

\[z^2 + z + \frac{1}{4} + \dfrac{3}{4}\]\[\left(z + \frac{1}{2}\right)^2 + \frac{3}{4}\]From this, it's clear that the answer is 3/4 because \(z\ne -1/2\).