# If $P$ is a real quadratic polynomial, then $|P(z)| \geq \sqrt{\Delta} |Im(z)|$

I was reading an article which contains a statement about the distance in the hyperbolic plane: if $$\gamma \in SL_2(\mathbb R)$$ is hyperbolic then $$\inf_{z \in H} d(\gamma z, z) = arcosh(Tr(\gamma)^2/2-1)$$. This implies the following:

If $$P$$ is a real quadratic polynomial with real roots and discriminant $$\Delta$$, then $$\inf_{z \in \mathbb C - \mathbb R} |P(z)|/|Im(z)| = \sqrt{\Delta}$$.

I suppose I could prove this by doing a long computation if I wanted to. But I'd like to know if there is a conceputal proof that explains where the discriminant comes from.