for the equations to have real roots either one of p2-4q or r2 - 4s must be greater than or equal to zero.
add these two: p2-4q + r2 - 4s = p2 + r2 - 4(q+ s) = p2 + r2 - 2pr (since pr = 2(q+s))
this is (p-r)2>=0 since perfect square.
so the sum is greater than zero hence one of them must be greater than zero and hence the either one of the equations has real roots.