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

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

Class 11 Mathematics 4-2.Quadratic Equations and Inequations Relation between roots and coefficients

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

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A
$r=p q$
B
$p q^2=r$
C
$r^2=p q$
D
$p q r=1$

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4-2.Quadratic Equations and Inequations

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Relation between roots and coefficients

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