Animated Solution for Mathematics - Quadratic Equations: If $\alpha$ and $\beta$ are the roots of the equation, $7x^2-3x-2=0$, then the value of $\frac{\alpha }{1-{\alpha }^2}+\frac{\beta }{1-{\beta }^2}$ is equal to :

Imagine you are standing before the quadratic equation $7x^2-3x-2=0$. A typical student might immediately reach for the quadratic formula, but as an elite JEE aspirant, you know that the roots are the DNA of the equation itself.

By comparing our equation to the standard form $a{x}^2+bx+c=0$, we identify the coefficients: $a=7$, $b=-3$, and $c=-2$.

Instead of hunting for the roots, we use the elegant tools provided by Vieta. The sum and product of the roots are defined as:

These two values are all we need. We do not care what $\alpha$ or $\beta$ are individually; we only care about how they behave together, which is the essence of symmetric functions.

We are tasked with evaluating the expression:

By combining the fractions using the LCM method, the numerator becomes $\alpha (1-{\beta }^2)+\beta (1-{\alpha }^2)$ and the denominator becomes $(1-{\alpha }^2)(1-{\beta }^2)$.

Expanding the numerator gives $\alpha -\alpha {\beta }^2+\beta -\beta {\alpha }^2$. Grouping these terms yields:

Now, we expand the denominator:

To solve this, we use the identity ${\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2-2\alpha \beta$.

First, we calculate the numerator:

Next, we calculate the denominator:

Finally, we divide the numerator by the denominator:

The $49$s cancel out, leaving us with the glorious result: $\frac{27}{16}$. This problem demonstrates that by trusting the identities, complexity dissolves into simplicity.