Animated Solution for Mathematics - Quadratic Equations: 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 :

Imagine you are standing before a quadratic equation, $6x^2-11x+\alpha =0$. Within its structure lies a profound question: for which values of the parameter $\alpha$ does this equation yield rational roots?

To solve this, we must look at the quadratic formula:

Here, $a=6$, $b=-11$, and $c=\alpha$. The soul of this equation, which dictates the nature of its roots, is the discriminant, $D=b^2-4ac$.

The Gatekeeper of Rationality

The term $\sqrt{D}$ is the gatekeeper of rationality. If $D$ is a perfect square, $\sqrt{D}$ becomes an integer, and the entire expression for $x$ becomes a ratio of integers—a rational number.

If $D$ is not a perfect square, $\sqrt{D}$ remains an irrational surd, and the roots are irrational. Our mission is to ensure that the discriminant satisfies the following condition:

For the roots to be real, the discriminant must be non-negative: $D\ge 0$. This gives us the inequality:

This simplifies to $24\alpha \le 121$, or $\alpha \le 5.04$. Assuming $\alpha$ is a positive integer, our search space is limited to $\alpha \in \{1,2,3,4,5\}$.

Now, let us test our candidates with mathematical precision:

For $\alpha =1$: $D=121-24(1)=97$ (Not a perfect square). For $\alpha =2$: $D=121-24(2)=73$ (Not a perfect square). For $\alpha =3$: $D=121-24(3)=49=7^2$ (Success). For $\alpha =4$: $D=121-24(4)=25=5^2$ (Success). For $\alpha =5$: $D=121-24(5)=1=1^2$ (Success*).

The Beauty of Method

We have found that $\alpha =3,4,5$ are the only values that satisfy the condition for rational roots. There are exactly three such values.

This journey demonstrates that even in complex problems, the path to the solution is paved with systematic, logical steps. By identifying the condition for rationality, constraining our search space, and testing our candidates, we have mastered the equation.