When you have a quadratic equation of the form $ax^2 + bx + c =0$, Vieta's formulas tell you that the roots, $x_1$ and $x_2$, are such that $x_1 + x_2 = \dfrac{-b}{a}$, and $x_1x_2 = \dfrac{c}{a}$.

Use these to rewrite $x_1^2 + x_2^2$ in terms of $a$, $b$, and $c$ only.