When the arguments of two roots of a quadratic equation are equal? Let $az^2+bz+c=0$ be a quadratic equation with complex coefficients $a,b,c$ and roots $z_1, z_2.$  
How can I obtain the condition for $$\arg z_1=\arg z_2$$ containing $a,b,c?$
At present I have,   Since $$z_1z_2=\dfrac{c}{a}$$ If $\arg z_1=\arg z_2,$
 then $$\arg z_1=\arg z_2=\dfrac{1}{2}(\arg c-\arg a)$$
Is there any reference discuss about roots of quadratic equations with complex coefficients?
Here, I have ask a similar question about $|z_1|=|z_2|,$ which has a nice solution. 

The roots $(-b\pm\sqrt{b^2-4ac})/2a$ are a real positive multiples of each other.
For some positive $\mu$,
$$-b+\sqrt{b^2-4ac}=\mu(-b-\sqrt{b^2-4ac})$$
$$(\mu-1)^2b^2=(\mu+1)^2(b^2-4ac)$$
$$b^2=\frac{(\mu+1)^2}{\mu}ac$$
The latter function of $\mu$ can take any value not smaller than $4$, and the condition is
$$b^2=\lambda ac,$$with $\lambda\ge4$, which can be expressed as $2\arg b=\arg a+\arg c\land |b|^2\ge4|ac|$.
