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If $r$ and $s$ are roots of $x^{2}+p x+q=0$, then what is the value $\operatorname{of}\left(1 / r^{2}\right)+\left(1 / s^{2}\right) ?$
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The correct answer is:
$\frac{p^{2}-2 q}{q^{2}}$
Since $r$ and $s$ are the roots of $x^{2}+p x+q=0$.
Then, $r+s=-p$ and $r s=q$ Now, $\frac{1}{r^{2}}+\frac{1}{s^{2}}=\frac{r^{2}+s^{2}}{(r s)^{2}}=\frac{(r+s)^{2}-2 r s}{(r s)^{2}}$
$=\frac{(-p)^{2}-2 q}{q^{2}}=\frac{p^{2}-2 q}{q^{2}}$
Then, $r+s=-p$ and $r s=q$ Now, $\frac{1}{r^{2}}+\frac{1}{s^{2}}=\frac{r^{2}+s^{2}}{(r s)^{2}}=\frac{(r+s)^{2}-2 r s}{(r s)^{2}}$
$=\frac{(-p)^{2}-2 q}{q^{2}}=\frac{p^{2}-2 q}{q^{2}}$
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