Let $\omega$ be a complex number such that $|\omega| = 1,$ and the equation
\[z^2 + z + \omega = 0\]has a pure imaginary root $z.$  Find $\omega + \overline{\omega}.$
Question:
Let $\omega$ be a complex number such that $|\omega| = 1,$ and the equation
\[z^2 + z + \omega = 0\]has a pure imaginary root $z.$  Find $\omega + \overline{\omega}.$
Answer:
Let's think step by step.

user
\sqrt{5} - 1
Let $\omega$ be a complex number such that $|\omega| = 1,$ and the equation
\[z^2 + z + \omega = 0\]has a pure imaginary root $z.$  Find $\omega + \overline{\omega}.$
train