MathNet 2014 Question 10

Let $x_1$ and $x_2$ be two different roots of the polynomial $p(x) = x^2 + a x + b$, and let $x_1^2 - \frac{1}{2}$ and $x_2^2 - \frac{1}{2}$ be the roots of $q(x) = x^2 + (a^2 - \frac{1}{2}) x + b^2 - \frac{1}{2}$. Find $a$ and $b$.