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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.

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