Prove that the equation x²+2px+2q=0 cannot have rational roots, if p and q are odd integers.

Question

Arun · Answer

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

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Prove that the equation x²+2px+2q=0 cannot have rational roots, if p and q are odd integers.

Prove that the equation x²+2px+2q=0 cannot have rational roots, if p and q are odd integers.

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