If the sum of any two roots of the equation x | vedclass

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

Question

(A) Let the roots of the equation $x^3+p x^2+q x+r=0$ be $\alpha, \beta, 	ext{ and } \gamma$.From the properties of roots,we have $\alpha+\beta+\gamm

Vedclass · Accepted Answer

$r=p q$

Vedclass · Answer

(A) Let the roots of the equation $x^3+p x^2+q x+r=0$ be $\alpha, \beta, 	ext{ and } \gamma$.From the properties of roots,we have $\alpha+\beta+\gamma = -p$ $(i)$.Given that the sum of two roots is zero,let $\beta+\gamma=0$.Substituting this into $(i)$,we get $\alpha = -p$.Since $\alpha$ is a root of the equation,it must satisfy $x^3+p x^2+q x+r=0$.Substituting $x = -p$:$(-p)^3 + p(-p)^2 + q(-p) + r = 0$$-p^3 + p^3 - p q + r = 0$$-p q + r = 0$$r = p q$.

If the sum of any two roots of the equation $x^3+p x^2+q x+r=0$ is zero,then

Similar Questions

If $\alpha, \beta$ are the roots of the equation $x^2+bx+c=0$ and $\alpha+h, \beta+h$ are the roots of the equation $x^2+qx+r=0$,then $h$ is equal to

If the arithmetic mean and geometric mean of the two roots of a quadratic equation are $9$ and $4$ respectively,then what is the quadratic equation?

If $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^{2}+bx+c=0$ and $3b^{2}=16ac$, then:

The cubic equation whose roots are thrice to each of the roots of $x^3+2x^2-4x+1=0$ is

What is the geometric mean of the roots of the equation $x^2 - 18x + 9 = 0$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module