If alpha and beta are the roots of the quadra | vedclass

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Question

(A) Given that $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^{2} + bx + c = 0$.From the properties of roots,we have the sum of root

Vedclass · Accepted Answer

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

Vedclass · Answer

(A) Given that $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^{2} + bx + c = 0$.From the properties of roots,we have the sum of roots $\alpha + \beta = -\frac{b}{a}$ and the product of roots $\alpha\beta = \frac{c}{a}$.Now,we need to find the value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$.Taking the common denominator,we get $\frac{\alpha}{\beta} + \frac{\beta}{\alpha} = \frac{\alpha^{2} + \beta^{2}}{\alpha\beta}$.Using the identity $\alpha^{2} + \beta^{2} = (\alpha + \beta)^{2} - 2\alpha\beta$,we substitute the values:$\frac{\alpha^{2} + \beta^{2}}{\alpha\beta} = \frac{(\alpha + \beta)^{2} - 2\alpha\beta}{\alpha\beta} = \frac{(-\frac{b}{a})^{2} - 2(\frac{c}{a})}{\frac{c}{a}}$.Simplifying the expression:$= \frac{\frac{b^{2}}{a^{2}} - \frac{2c}{a}}{\frac{c}{a}} = \frac{\frac{b^{2} - 2ac}{a^{2}}}{\frac{c}{a}} = \frac{b^{2} - 2ac}{a^{2}} 	imes \frac{a}{c} = \frac{b^{2} - 2ac}{ac}$.

If $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^{2} + bx + c = 0$,then the value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$ is

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