Question

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

Answer 1

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

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

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

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

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

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

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