Question

The number of all possible positive integral values of \(\alpha\) for which the roots of the quadratic equation, \(6x^2-11x+\alpha=0\) are rational numbers is :

Answer 1

\(6x^2-11x+\alpha=0\)

If roots are rational numbers, then discriminant should be a perfect square.

\(D=121-24\alpha\)

So, If \(\alpha=1\), \(D=97\), which is not a perfect square.

\(\alpha=2\), \(D=73\), which is not a perfect square.

\(\alpha=3\), \(D=49\), it is a perfect square.

\(\alpha=4\), \(D=25\), it is a perfect square.

\(\alpha=5\), \(D=1\), it is a perfect square.

So, number of possible values of \(\alpha\) is \(3\).