The value of a for which the equations x^3+ax | vedclass

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

Question

(A) Let the common root be $\alpha$. Then $\alpha^3 + a\alpha + 1 = 0$ and $\alpha^4 + a\alpha^2 + 1 = 0$.From the first equation,$a\alpha = -\alpha^3

Vedclass · Accepted Answer

-$2$

Vedclass · Answer

(A) Let the common root be $\alpha$. Then $\alpha^3 + a\alpha + 1 = 0$ and $\alpha^4 + a\alpha^2 + 1 = 0$.From the first equation,$a\alpha = -\alpha^3 - 1$.Substituting this into the second equation: $\alpha^4 + \alpha(a\alpha^2) + 1 = 0$ is not direct,so multiply the first by $\alpha$: $\alpha^4 + a\alpha^2 + \alpha = 0$.Subtracting this from the second equation: $(\alpha^4 + a\alpha^2 + 1) - (\alpha^4 + a\alpha^2 + \alpha) = 0$.This gives $1 - \alpha = 0$,so $\alpha = 1$.Substituting $\alpha = 1$ into the first equation: $1^3 + a(1) + 1 = 0$.$1 + a + 1 = 0$,which implies $a = -2$.

The value of $a$ for which the equations $x^3+ax+1=0$ and $x^4+ax^2+1=0$ have a common root is

Similar Questions

The value of $a$ for which the equations $x^2 - 3x + a = 0$ and $x^2 + ax - 3 = 0$ have a common root is

Two distinct polynomials $f(x)$ and $g(x)$ are defined as follows: $f(x)=x^2+ax+2$ and $g(x)=x^2+2x+a$. If the equations $f(x)=0$ and $g(x)=0$ have a common root,then the sum of the roots of the equation $f(x)+g(x)=0$ is:

If $\alpha, \beta$ and $\gamma$ are three consecutive terms of a non-constant $G.P.$ such that the equations $\alpha x^2 + 2\beta x + \gamma = 0$ and $x^2 + x - 1 = 0$ have a common root,then $\alpha(\beta + \gamma)$ is equal to

If a root of the equations $x^2 + px + q = 0$ and $x^2 + \alpha x + \beta = 0$ is common,then its value will be (where $p \neq \alpha$ and $q \neq \beta$)

If the roots of the equations $x^2 + 3x + 2 = 0$ and $x^2 - x + \lambda = 0$ are in the same ratio,then what is the value of $\lambda$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module