MathNet 2014 Question 49

Let $x_1,x_2$ be distinct zeros of $p(x)=x^2+ax+b$, and suppose $x_1^2-\frac12$ and $x_2^2-\frac12$ are zeros of $q(x)=x^2+(a^2-\frac12)x+b^2-\frac12$. Determine $a,b$.