For the quadratic equation ax^2 + bx + c = 0, | vedclass

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(D) Given the roots $\alpha$ and $\beta$ for $ax^2 + bx + c = 0$,we have $\alpha + \beta = -\frac{b}{a}$ and $\alpha\beta = \frac{c}{a}$.Consider the

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$-\frac{2}{a}$

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(D) Given the roots $\alpha$ and $\beta$ for $ax^2 + bx + c = 0$,we have $\alpha + \beta = -\frac{b}{a}$ and $\alpha\beta = \frac{c}{a}$.Consider the expression $E = \frac{\alpha}{a\beta + b} + \frac{\beta}{a\alpha + b}$.Since $a\alpha^2 + b\alpha + c = 0$,we have $a\alpha + b = -\frac{c}{\alpha}$. Similarly,$a\beta + b = -\frac{c}{\beta}$.Substituting these into the expression:$E = \frac{\alpha}{-c/\beta} + \frac{\beta}{-c/\alpha} = -\frac{\alpha\beta}{c} - \frac{\alpha\beta}{c} = -\frac{2\alpha\beta}{c}$.Substituting $\alpha\beta = \frac{c}{a}$:$E = -\frac{2(c/a)}{c} = -\frac{2}{a}$.

For the quadratic equation $ax^2 + bx + c = 0$,if $\alpha$ and $\beta$ are the roots,then $\frac{\alpha}{a\beta + b} + \frac{\beta}{a\alpha + b} = \dots$

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