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If $\alpha, \beta$ are the roots of the equation $a x^{2}+b x+b=0$, then
what is the value of $\sqrt{\frac{\alpha}{o}}+\sqrt{\frac{\beta}{\alpha}}+\sqrt{\underline{b}} ?$
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what is the value of $\sqrt{\frac{\alpha}{o}}+\sqrt{\frac{\beta}{\alpha}}+\sqrt{\underline{b}} ?$
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$\alpha$ and $\beta$ are the roots of the given equation, then
$\alpha+\beta=-\frac{\mathrm{b}}{\mathrm{a}}$ and $\alpha \beta=\frac{\mathrm{b}}{\mathrm{a}}$
$\sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\beta}{\alpha}}+\sqrt{\frac{\mathrm{b}}{\mathrm{a}}}$
$=\frac{\alpha+\beta}{\sqrt{\alpha \beta}}+\sqrt{\alpha \beta}$
$=\frac{\alpha+\beta+\alpha \beta}{\sqrt{\alpha \beta}}=\frac{-\frac{\mathrm{b}}{\mathrm{a}}+\frac{\mathrm{b}}{\mathrm{a}}}{\sqrt{\frac{\mathrm{b}}{\mathrm{a}}}}=0$
$\alpha+\beta=-\frac{\mathrm{b}}{\mathrm{a}}$ and $\alpha \beta=\frac{\mathrm{b}}{\mathrm{a}}$
$\sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\beta}{\alpha}}+\sqrt{\frac{\mathrm{b}}{\mathrm{a}}}$
$=\frac{\alpha+\beta}{\sqrt{\alpha \beta}}+\sqrt{\alpha \beta}$
$=\frac{\alpha+\beta+\alpha \beta}{\sqrt{\alpha \beta}}=\frac{-\frac{\mathrm{b}}{\mathrm{a}}+\frac{\mathrm{b}}{\mathrm{a}}}{\sqrt{\frac{\mathrm{b}}{\mathrm{a}}}}=0$
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