2011/083) Show that for p and q odd x^2 + 2px + 2q = 0 does not have rational solution

proof

the discriminant is

b^2-4ac = (2p)^2 – 8q = 4(p^2-2q)

as p and q are odd

p^2-2q = (2s+1)^2 – 2(2t+1) = (4s^2+ 4s+1) – (4t+2) =(4s^2 + 4s – 4t -1) which is 3 mod 4

so discriminant cannot be a perfect square so there is no rational root