Let $a$, $b$, $c$ be positive integers with $a \le 10$.  Suppose the parabola $y = ax^2 + bx + c$ meets the $x$-axis at two distinct points $A$ and $B$.  Given that the length of $\overline{AB}$ is irrational, determine, with proof, the smallest possible value of this length, across all such choices of $(a, b, c)$.