Problem 5

The equation $x^2 + bx + c = 0$ has two different integer solutions, and the equation $x^2 + bx - c = 0$ also has two different integer solutions, where $b$ and $c$ are nonzero.

(i) Show that it is possible to find different positive integers $p$ and $q$ such that $2b^2 = p^2 + q^2$.

(ii) Show that it is possible to find different positive integers $r$ and $s$ such that $b^2 = r^2 + s^2$.