The values of a and b for which the equation | vedclass

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

Question

(D) Let the four real roots be $\alpha, \beta, \gamma, \delta$. The equation is $(x - \alpha)(x - \beta)(x - \gamma)(x - \delta) = 0$.Comparing this w

Vedclass · Accepted Answer

$a = 6, b = -4$

Vedclass · Answer

(D) Let the four real roots be $\alpha, \beta, \gamma, \delta$. The equation is $(x - \alpha)(x - \beta)(x - \gamma)(x - \delta) = 0$.Comparing this with $x^4 - 4x^3 + ax^2 + bx + 1 = 0$,we get:$\sum \alpha = 4$,$\sum \alpha\beta = a$,$\sum \alpha\beta\gamma = -b$,and $\alpha\beta\gamma\delta = 1$.For real positive roots,the Arithmetic Mean $(AM)$ $\ge$ Geometric Mean $(GM)$.$AM = \frac{\alpha + \beta + \gamma + \delta}{4} = \frac{4}{4} = 1$.$GM = (\alpha\beta\gamma\delta)^{1/4} = (1)^{1/4} = 1$.Since $AM = GM = 1$,all roots must be equal,i.e.,$\alpha = \beta = \gamma = \delta = 1$.Now,$a = \sum \alpha\beta = \binom{4}{2} 	imes (1 	imes 1) = 6 	imes 1 = 6$.$-b = \sum \alpha\beta\gamma = \binom{4}{3} 	imes (1 	imes 1 	imes 1) = 4 	imes 1 = 4$,so $b = -4$.Thus,$a = 6$ and $b = -4$.

The values of $a$ and $b$ for which the equation $x^4 - 4x^3 + ax^2 + bx + 1 = 0$ has four real roots are:

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 $\lambda \in R$ is such that the sum of the cubes of the roots of the equation $x^2 + (2 - \lambda) x + (10 - \lambda) = 0$ is minimum,then the magnitude of the difference of the roots of this equation is

The number of solutions of $\log_4(x - 1) = \log_2(x - 3)$ is:

If the graph of $y = ax^2 - bx + c$ is as shown below,then the signs of $a$,$b$,and $c$ are:

If $\alpha$ and $\beta$ are roots of the equation $x^2 - 4\sqrt{2}kx + 2e^{4\ln k} - 1 = 0$ for some $k$, and $\alpha^2 + \beta^2 = 66$, then $\alpha^3 + \beta^3$ is equal to: (in $\sqrt{2}$)

Vedclass Test Series

Exam Paper Generator

Online Exam Module