So we require that $p^{2}-4q > 0, r^{2}-4s > 0$ It's pretty clear that $k \ge 4$. Let $p, r$ be the two larger values and $q, s$ be the two smaller values. Then both discriminants are guaranteed to be positive, and we only need to guarantee that the roots are distinct. If, given our above arrangement, there exists a repeated root, switch $q$ and $s$. (I can't prove that this will get rid of the repeated roots, but it looks like annoying casework to me with too many variables )