Coefficient and root

The answer is when $a,b,c$ are all odd integers.

$PROOF:$

Let $x = \frac{p}{q}$ be a rational root ( $p,q \in \mathbb{Z}, q \neq 0,$ and $p, q$ both not even). This in turn gives us:

$ap^2 + bpq + cq^2 = 0$ .

If $p, q =$ odd AND $a,b,c =$ odd, then we have a contradiction since the sum of three odd integers can never equal zero. If $p =$ even (odd), $q =$ odd (even) AND $a,b,c =$ odd, then we have the sum of two even integers & one odd integer (which is also odd and a contradiction).

Therefore, no rational roots exist when $a,b,c$ are all odd integers.

$\mathbb{Q.} \mathbb{E.} \mathbb{D.}$