If alpha and beta are roots of the equation A | vedclass

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Question

(A) Given the quadratic equation $Ax^2 + Bx + C = 0$.From the relationship between roots and coefficients,we have:Sum of roots: $\alpha + \beta = -\fr

Vedclass · Accepted Answer

$\frac{3ABC - B^3}{A^3}$

Vedclass · Answer

(A) Given the quadratic equation $Ax^2 + Bx + C = 0$.From the relationship between roots and coefficients,we have:Sum of roots: $\alpha + \beta = -\frac{B}{A}$Product of roots: $\alpha \beta = \frac{C}{A}$We use the algebraic identity: $\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha \beta (\alpha + \beta)$.Substituting the values:$\alpha^3 + \beta^3 = \left(-\frac{B}{A}\right)^3 - 3\left(\frac{C}{A}\right)\left(-\frac{B}{A}\right)$$= -\frac{B^3}{A^3} + \frac{3BC}{A^2}$To combine these,find a common denominator $A^3$:$= \frac{-B^3 + 3ABC}{A^3} = \frac{3ABC - B^3}{A^3}$.

If $\alpha$ and $\beta$ are roots of the equation $Ax^2 + Bx + C = 0$,then the value of $\alpha^3 + \beta^3$ is

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