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 $x_1$ and $x_2$ are the real roots of the equation $x^2-kx+c=0$,then the distance between the points $A(x_1, 0)$ and $B(x_2, 0)$ is

If the equation having the roots as the values obtained by diminishing each root of the equation $x^3-3x^2+2x-1=0$ by $K$ is $x^3-x-1=0$,then $K=$

If $\alpha$ and $\beta$ are roots of the equation $Ax^2 + Bx + C = 0$,then the value of $\alpha^3 + \beta^3$ is

If the ratio of the roots of $x^2 + bx + c = 0$ and $x^2 + qx + r = 0$ is the same,then

If the roots of the quadratic equation $x^2 + px + q = 0$ are $\tan 30^\circ$ and $\tan 15^\circ$ respectively,then the value of $2 + q - p$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module