Question

If the roots of $7x^{2}-15x+\alpha =0$ are rational numbers, then the number of all possible positive integral values of $\alpha $ is,

Answer 1

since, the roots of $7x^{2}-15x+\alpha =0$ are rational numbers.
$\Rightarrow $ its discriminant must be a perfect square.
Discriminant $=(-15-4(7)(\alpha)$
$=225-28\alpha $
For $225-28\alpha $ to be a perfect square, the possible values of $\alpha $ are 2 and 8.
$\Rightarrow $ Number of positive integral values of $\alpha =2$