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

Vedclass · Accepted Answer

$\frac{b^2 + 3ac}{a^2}$

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(A) For the quadratic equation $ax^2 - bx - c = 0$,the sum of the roots is $\alpha + \beta = -(\frac{-b}{a}) = \frac{b}{a}$ and the product of the roots is $\alpha\beta = \frac{-c}{a}$.We need to find the value of $\alpha^2 - \alpha\beta + \beta^2$.This can be rewritten as $(\alpha + \beta)^2 - 2\alpha\beta - \alpha\beta = (\alpha + \beta)^2 - 3\alpha\beta$.Substituting the values of $(\alpha + \beta)$ and $\alpha\beta$:$= (\frac{b}{a})^2 - 3(\frac{-c}{a})$$= \frac{b^2}{a^2} + \frac{3c}{a}$$= \frac{b^2 + 3ac}{a^2}$.

If $\alpha$ and $\beta$ are the roots of the equation $ax^2 - bx - c = 0$,then $\alpha^2 - \alpha\beta + \beta^2 = .......$

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