If alpha and beta are roots of ax^2 + 2bx + c | vedclass

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Question

(C) Given the quadratic equation $ax^2 + 2bx + c = 0$.From the properties of roots,the sum of roots $\alpha + \beta = -\frac{2b}{a}$ and the product o

Vedclass · Accepted Answer

$-\frac{2b}{\sqrt{ac}}$

Vedclass · Answer

(C) Given the quadratic equation $ax^2 + 2bx + c = 0$.From the properties of roots,the sum of roots $\alpha + \beta = -\frac{2b}{a}$ and the product of roots $\alpha \beta = \frac{c}{a}$.We need to evaluate the expression $\sqrt{\frac{\alpha}{\beta}} + \sqrt{\frac{\beta}{\alpha}}$.Taking the common denominator,we get $\frac{\sqrt{\alpha}}{\sqrt{\beta}} + \frac{\sqrt{\beta}}{\sqrt{\alpha}} = \frac{(\sqrt{\alpha})^2 + (\sqrt{\beta})^2}{\sqrt{\alpha \beta}} = \frac{\alpha + \beta}{\sqrt{\alpha \beta}}$.Substituting the values of $\alpha + \beta$ and $\alpha \beta$:$= \frac{-2b/a}{\sqrt{c/a}} = \frac{-2b/a}{\sqrt{c}/\sqrt{a}} = -\frac{2b}{a} 	imes \frac{\sqrt{a}}{\sqrt{c}} = -\frac{2b}{\sqrt{a} \cdot \sqrt{a}} 	imes \frac{\sqrt{a}}{\sqrt{c}} = -\frac{2b}{\sqrt{ac}}$.

If $\alpha$ and $\beta$ are roots of $ax^2 + 2bx + c = 0$,then $\sqrt{\frac{\alpha}{\beta}} + \sqrt{\frac{\beta}{\alpha}}$ is equal to

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