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) Given the quadratic equation $x^2 + ax + b = 0$.By the relation between roots and coefficients,the sum of roots $\alpha + \beta = -a$ and the prod

Vedclass · Accepted Answer

$- a^3 + 3ab$

Vedclass · Answer

(C) Given the quadratic equation $x^2 + ax + b = 0$.By the relation between roots and coefficients,the sum of roots $\alpha + \beta = -a$ and the product of roots $\alpha \beta = b$.We know the algebraic identity $\alpha^3 + \beta^3 = (\alpha + \beta)(\alpha^2 - \alpha \beta + \beta^2)$.This can be rewritten as $\alpha^3 + \beta^3 = (\alpha + \beta)[(\alpha + \beta)^2 - 3\alpha \beta]$.Substituting the values,we get $\alpha^3 + \beta^3 = (-a)[(-a)^2 - 3(b)]$.$\alpha^3 + \beta^3 = (-a)(a^2 - 3b) = -a^3 + 3ab$.

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

Let $\alpha, \beta$ be the roots of the equation $x^2-px+r=0$ and $\frac{\alpha}{2}, 2\beta$ be the roots of the equation $x^2-qx+r=0$. Then the value of $r$ is

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+p x^2+q x+r=0$,then $(\alpha+\beta)(\beta+\gamma)(\gamma+\alpha) =$

If $\alpha, \beta$ are the roots of $x^2 + px + 1 = 0$ and $\gamma, \delta$ are the roots of $x^2 + qx + 1 = 0$,then $(\alpha - \gamma) (\beta - \gamma) (\alpha + \delta) (\beta + \delta) = \dots$

If the roots of the equation $5x^2 - 7x + k = 0$ are reciprocal to each other,then the value of $k$ is:

In a triangle $PQR$,$\angle R = \frac{\pi}{2}$. If $\tan(\frac{P}{2})$ and $\tan(\frac{Q}{2})$ are the roots of the equation $ax^2 + bx + c = 0$ $(a \neq 0)$,then:

Vedclass Test Series

Exam Paper Generator

Online Exam Module