HMMT 2015 U9

Let $a, b, c$ be integers. Define $f(x)=a x^{2}+b x+c$. Suppose there exist pairwise distinct integers $u, v, w$ such that $f(u)=0, f(v)=0$, and $f(w)=2$. Find the maximum possible value of the discriminant $b^{2}-4 a c$ of $f$.