Question

Let $\alpha ,\beta$ are the roots of the quadratic equation $a{x}^2+bx+c=0$ then find the value of $\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}$.

Answer 1

Given $\alpha ,\beta$ are the roots of the quadratic equation $a{x}^2+bx+c=0$.

Then from the relation between the roots and the co-efficients we get,

$\alpha +\beta =-\frac{b}{a}$.......(1) and $\alpha .\beta =\frac{c}{a}$......(2).

Now,

$\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}$

$=\frac{{\alpha }^2+{\beta }^2}{(\alpha .\beta {)}^2}$

$=\frac{(\alpha +\beta {)}^2-2\alpha .\beta }{(\alpha .\beta {)}^2}$

$=\frac{\frac{b^2}{a^2}-2.\frac{c}{a}}{\frac{c^2}{a^2}}$[ Using (1) and (2)]

$=\frac{b^2-2ac}{c^2}$.