Using Quadratic formula, solve the following quadratic equation for $x$ .
$p^{2} x^{2} + (p^{2} - q^{2}) x - q^{2} = 0$

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Solution

$\begin{matrix} p^{2} x^{2} + (p^{2} - q^{2}) x - q^{2} = 0 = \frac{- (p^{2} - q^{2}) + \sqrt{{(p^{2} - q^{2})}^{2} + 4 p^{2} q^{2}}}{2 p^{2}} = \frac{q^{2} - p^{2} \pm (p^{2} + q^{2})}{2 p^{2}} \Rightarrow x = \frac{q^{2}}{p^{2}} a n d x = - 1 \end{matrix}$

Hence, this is the answer.

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