If alpha, beta, gamma are the roots of the eq | vedclass

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

Question

(D) Given the cubic equation $x^3-5x^2-2x+24=0$,the roots are $\alpha, \beta, \gamma$. From Vieta's formulas: $\alpha+\beta+\gamma = 5$ $\alpha\beta+\

Vedclass · Accepted Answer

$\frac{-61}{6}$

Vedclass · Answer

(D) Given the cubic equation $x^3-5x^2-2x+24=0$,the roots are $\alpha, \beta, \gamma$. From Vieta's formulas: $\alpha+\beta+\gamma = 5$ $\alpha\beta+\beta\gamma+\gamma\alpha = -2$ $\alpha\beta\gamma = -24$ We need to evaluate the expression: $E = \frac{\beta\gamma}{\alpha}+\frac{\gamma\alpha}{\beta}+\frac{\alpha\beta}{\gamma}$ $E = \frac{(\beta\gamma)^2 + (\gamma\alpha)^2 + (\alpha\beta)^2}{\alpha\beta\gamma}$ Using the identity $a^2+b^2+c^2 = (a+b+c)^2 - 2(ab+bc+ca)$,where $a=\alpha\beta, b=\beta\gamma, c=\gamma\alpha$: $E = \frac{(\alpha\beta+\beta\gamma+\gamma\alpha)^2 - 2(\alpha\beta^2\gamma + \beta\gamma^2\alpha + \gamma\alpha^2\beta)}{\alpha\beta\gamma}$ $E = \frac{(\alpha\beta+\beta\gamma+\gamma\alpha)^2 - 2\alpha\beta\gamma(\beta+\gamma+\alpha)}{\alpha\beta\gamma}$ Substituting the values: $E = \frac{(-2)^2 - 2(-24)(5)}{-24}$ $E = \frac{4 + 240}{-24} = \frac{244}{-24} = -\frac{61}{6}$

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-5x^2-2x+24=0$,then $\frac{\beta\gamma}{\alpha}+\frac{\gamma\alpha}{\beta}+\frac{\alpha\beta}{\gamma}=$

Similar Questions

If ${x^2} + px + q = 0$ is the quadratic equation whose roots are $a - 2$ and $b - 2$,where $a$ and $b$ are the roots of ${x^2} - 3x + 1 = 0$,then:

If $\frac{1}{\sqrt{\alpha}}$ and $\frac{1}{\sqrt{\beta}}$ are the roots of the equation $ax^2 + bx + 1 = 0$ $(a \ne 0, a, b \in R)$,then the equation $x(x + b^3) + (a^3 - 3abx) = 0$ has roots

If $2$ and $3$ are two roots of the equation $2x^3 + mx^2 - 13x + n = 0$,then the values of $m$ and $n$ are respectively

If the sum of the squares of the roots of the equation $x^2-(a-2)x-(a+1)=0$ is least for an appropriate value of the variable parameter $a$, then the value of $a$ will be

If $\alpha, \beta, \gamma$ are the roots of $x^3 - x - 2 = 0$,then the value of $\alpha^5 + \beta^5 + \gamma^5$ is-

Vedclass Test Series

Exam Paper Generator

Online Exam Module