If $α$ and $β$ the zeros of the quadratic polynomial $a x^{2} + b x + c = 0$ then find the value $\frac{1}{α^{2}} + \frac{1}{β^{2}}$

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Solution

$\frac{1}{α^{2}} + \frac{1}{β^{2}}$

$= \frac{α^{2} + β^{2}}{α^{2} β^{2}}$

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

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

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

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