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Q.
Find the number of all possible positive integral values of $\alpha$ for which the roots of $7 x^{2}-15 x+\alpha=0$ are rational numbers.
Complex Numbers and Quadratic Equations
Solution:
Since, the roots of $7 x^{2}-15 x+\alpha=0$ 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 $\alpha=2$.