For an algebraic number $\alpha$ of degree $2$, I know how to prove that there is a constant $c>0$ such that $$\left|\alpha - \frac{p}{q}\right| \geq \frac{c}{q^2}.$$
However, I get the constant $c$ from the Taylor expansion of the minimal polynomial at $\alpha$. Apparently it is possible to relate $c$ to the discriminant of the polynomial, how can I do it?
Also, is there something known about the best possible $c$ in more general cases? Is there also an expression in terms of discriminant-like quantity linked to the minimal polynomial?