Prove $ax^2+bx+c=0$ has no rational roots if $a,b,c$ are odd If $a,b,c$ are odd, how can we prove that $ax^2+bx+c=0$ has no rational roots?
I was unable to proceed beyond this: Roots are $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
and rational numbers are of the form $\frac pq$.

If $\frac{p}{q}$ is a root then $a\frac{p^2}{q^2}+b\frac{p}{q}+c=0\Rightarrow ap^2+bpq+cq^2=0$. Now we may assume that $p,q$ are not both even; $a,b,c$ are odd, whence contradiction ($p(ap+bq)=-cq^2$ with $p$ even $c,q$ odd or $q(bp+cq)=-ap^2$ with $q$ even $a,p$ odd or ...). 
