Question

Find the number of all possible positive integral values of $\alpha$ for which the roots of $7x^2-15x+\alpha =0$ are rational numbers.

Answer 1

Answer: 2

Since, the roots of $7x^2-15x+\alpha =0$ are rational numbers.
$\Rightarrow$ its discriminant must be a perfect square.
Discriminant $=(-15{)}^2-4(7)(\alpha )$
$=225-28\alpha$
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 $\alpha =2$.