Let $az^2+bz+c=0$ be a quadratic equation with complex coefficients $a,b,c$ and roots $z_1, z_2.$
If it is given that $|z_1|\not=|z_2|,$ how can I obtain the condition for this containing $a,b,c?$
Is there any reference discuss about roots of quadratic equations with complex coefficients?
Let us express that the magnitudes are equal. Using the classical formula:
$$\left|\frac{-b+\sqrt{b^2-4ac}}{2a}\right|=\left|\frac{-b-\sqrt{b^2-4ac}}{2a}\right|.$$
If $b=0$, the equality holds.
Otherwise, we can multiply by $|-2a/b|$, and
$$\left|1-\sqrt{1-4\frac{ac}{b^2}}\right|=\left|1+\sqrt{1-4\frac{ac}{b^2}}\right|.$$
Now, $|1-p|=|1+p|\implies \Re(p)=0$. Indeed, $(1-x)^2+y^2=(1+x)^2+y^2\implies x=0$.
So $$\sqrt{1-4\frac{ac}{b^2}}=iy,$$ $$ac=\frac{1-y^2}4b^2.$$
In conclusion, the magnitudes differ if $b$ is nonzero and $ac$ is not the product of $b^2$ by a real number less than or equal to $1/4$.