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

Vedclass · Accepted Answer

$\frac{c(a - b)}{a^2}$

Vedclass · Answer

(A) Given the quadratic equation $ax^2 + bx + c = 0$,the sum of the roots is $\alpha + \beta = -\frac{b}{a}$ and the product of the roots is $\alpha \beta = \frac{c}{a}$.We need to evaluate the expression $\alpha \beta^2 + \alpha^2 \beta + \alpha \beta$.Factoring out $\alpha \beta$,we get:$\alpha \beta (\beta + \alpha) + \alpha \beta = \alpha \beta (\alpha + \beta + 1)$.Substituting the values of $\alpha + \beta$ and $\alpha \beta$:$= \frac{c}{a} \left( -\frac{b}{a} + 1 \right) = \frac{c}{a} \left( \frac{a - b}{a} \right) = \frac{c(a - b)}{a^2}$.

If $\alpha$ and $\beta$ are the roots of the equation $ax^2 + bx + c = 0$,then what is the value of $\alpha \beta^2 + \alpha^2 \beta + \alpha \beta$?

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