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

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Solution

Given $α, β$ be the roots of the equation $a x^{2} + b x + c = 0$ , then
$α + β = - \frac{b}{a}$ .......(1) and $α . β = \frac{c}{a}$ ......(2).

Now $\frac{1}{α^{2}} + \frac{1}{β^{2}}$

$= \frac{α^{2} + β^{2}}{α^{2} . β^{2}}$

$= \frac{(α + β)^{2} - 2 α . β}{α^{2} . β^{2}}$

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

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

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