HMMT 2018 U29

Let $a, b, c$ be positive integers. All the roots of each of the quadratics

$$a x^{2}+b x+c, a x^{2}+b x-c, a x^{2}-b x+c, a x^{2}-b x-c$$

are integers. Over all triples $(a, b, c)$, find the triple with the third smallest value of $a+b+c$.