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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) For the quadratic equation $x^2 + ax + b = 0$,the sum of the roots is $\alpha + \beta = -a$ and the product of the roots is $\alpha\beta = b$.We u

Vedclass · Accepted Answer

$- a^3 + 3ab$

Vedclass · Answer

(C) For the quadratic equation $x^2 + ax + b = 0$,the sum of the roots is $\alpha + \beta = -a$ and the product of the roots is $\alpha\beta = b$.We use the algebraic identity for the sum of cubes: $\alpha^3 + \beta^3 = (\alpha + \beta)(\alpha^2 - \alpha\beta + \beta^2)$.This can be rewritten in terms of the sum and product of roots as: $\alpha^3 + \beta^3 = (\alpha + \beta)[(\alpha + \beta)^2 - 3\alpha\beta]$.Substituting the values $\alpha + \beta = -a$ and $\alpha\beta = b$:$\alpha^3 + \beta^3 = (-a)[(-a)^2 - 3(b)]$$= (-a)(a^2 - 3b)$$= -a^3 + 3ab$.

If $\alpha$ and $\beta$ are the roots of the equation $x^2 + ax + b = 0$,then the value of $\alpha^3 + \beta^3$ is equal to:

Similar Questions

Number of integral values of $m$ for which ${x}^2 + 5m{x} - 3m + 1 < 0$ $\forall x \in R$ is (where ${.}$ denotes the fractional part function).

The roots of the equation $ax^2 + bx + c = 0$ will be reciprocal if

Find the quadratic equation whose roots are reciprocals of the roots of the equation $3x^{2}-20x+17=0$.

The expression $x^{4}+7 x^{2}+16$ can be factored as:

The value of $k$ for which the equation $(K - 2)x^2 + 8x + K + 4 = 0$ has both roots real,distinct and negative is

Vedclass Test Series

Exam Paper Generator

Online Exam Module