If `a ,b ,a n dc` are odd integers, then prove that roots of `a x^2+b x+c=0` cannot be rational.

Explanation

Here, discriminant `D =b^(2) - 4ac.` Suppose the roots are rational. Thus , D will be a perfect square.
Let `b^(2) - 4 ac = d^(2)` . Since a, b and c are odd integers, d will be odd .Now,
`b^(2) - d^(2) = 4ac`
Let b = 2k + 1 and d = 2m + 1 . Then
`b^(2) - d^(2) = (b -d) (b + d)`
`2(k -m) 2(k + m + 1)`
Now, either (k - m) or (k + m + 1) is always even. Hence, `b^(2)-d^(2)` is always a multiple of 8. But , 4ac is a multiple of 4 (not of 8 ), Which is a contradiction. Hence, the roots of `ax^(2) + bx + c =0` connot be rational.