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If $\alpha$ and $\beta$ are the roots of the equation $x^{2}+b x+c=0$, then what is the value of $\alpha^{-1}+\beta^{-1} ?$
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The correct answer is:
$-\frac{b}{c}$
Consider $\alpha^{-1}+\beta^{-1}=\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha \beta}$
Equation is $x^{2}+b x+c=0$
Now, sum of roots $=\alpha+\beta=-b$
and product of roots $=\alpha \cdot \beta=\mathrm{c}$
$\frac{\alpha+\beta}{\alpha \beta}=-\frac{\mathrm{b}}{\mathrm{c}}$
Hence, $\alpha^{-1}+\beta^{-1}=-\frac{\mathrm{b}}{\mathrm{c}}$
Equation is $x^{2}+b x+c=0$
Now, sum of roots $=\alpha+\beta=-b$
and product of roots $=\alpha \cdot \beta=\mathrm{c}$
$\frac{\alpha+\beta}{\alpha \beta}=-\frac{\mathrm{b}}{\mathrm{c}}$
Hence, $\alpha^{-1}+\beta^{-1}=-\frac{\mathrm{b}}{\mathrm{c}}$
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