Let $a, b \in \mathbb{C}^{*}$ and $(E)$ the following equation : $z^2 - az + b = 0$.
Assuming : $\left\{ \begin{equation} \begin{aligned} & \arg b \equiv 2\arg a \pmod{2\pi} \\ & |a|^2 - 4|b| \geq 0 \end{aligned} \end{equation} \right. $
How should I start in order to show that $z_1$ and $z_2$ (roots of $(E)$) would have the same arguments ($\arg z_1 = \arg z_2 \pmod{2\pi}$).
I have proved the first implication by assuming $z_1$ and $z_2$ have the same arguments, hence the set of conditions above is true.
For the reciprocal, I tried to start again from $b = z_1 z_2$ and $a = z_1 + z_2$ and manipulating these equations, but I have difficulty in wrapping my mind around the simultaneous usage of these two conditions.
I wonder also if that's provable through counter example?