When does a cubic equation has two roots with same absolute values Let $$z^3+bz^2+cz+d=0$$ be a cubic equation with complex coefficients. Suppose $z_1, z_2$ and $z_3$ are its roots.
I need to find a condition on $b,c,d$ so that $$|z_1|=|z_2|.$$ How can I find such a relationship? Any help would be appreciate. Thank you.

In comparison to Robert Israel's answer, I note the following:
$$|ue^{\alpha\cdot i}|=u$$
for $\alpha\in\mathbb{R}$ and $u>0$.
Generally, Euler form of a complex number has the magnitude of the complex number as a coefficient and the real angle multiplied by $i$ with base $e$, which allows us to encompass all possible complex numbers with a slight exception of maybe $0+0i$.
Now, we can define our problem such that we have:
$$z_1=ue^{xi}, z_2=ue^{yi}$$
So now we have:
$$(z-ue^{xi})(z-ue^{yi})(z-z_3)$$
And you could multiply them all out if you want.
Another answer goes along the line of finding the root of a cubic polynomial, which may be found on Wolfram|Alpha.
Take the absolute value of the thing and try to solve using definitions of the absolute value operation:
$$|z|=|a+bi|=\sqrt{a^2+b^2}$$
Or:
$$|z|=z, z\ge0$$
$$|z|=-z,z\le0$$
