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

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(B) Given that $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^{2} + bx + c = 0$.According to the relationship between roots and coef

Vedclass · Accepted Answer

$\frac{b(3ac - b^{2})}{a^{3}}$

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(B) Given that $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^{2} + bx + c = 0$.According to the relationship between roots and coefficients:Sum of roots: $\alpha + \beta = -\frac{b}{a}$Product of roots: $\alpha\beta = \frac{c}{a}$We know the algebraic identity: $\alpha^{3} + \beta^{3} = (\alpha + \beta)^{3} - 3\alpha\beta(\alpha + \beta)$Substituting the values:$\alpha^{3} + \beta^{3} = (-\frac{b}{a})^{3} - 3(\frac{c}{a})(-\frac{b}{a})$$= -\frac{b^{3}}{a^{3}} + \frac{3bc}{a^{2}}$Taking $a^{3}$ as the common denominator:$= \frac{-b^{3} + 3abc}{a^{3}}$$= \frac{b(3ac - b^{2})}{a^{3}}$

If $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^{2} + bx + c = 0$,the value of $\alpha^{3} + \beta^{3}$ is

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