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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Aicharya · Accepted Answer

Given equation, $x^3+p x^2+q x+r$ Let roots of given equation are $\alpha, \beta$ and $\gamma$

[From Eqs. (i) and (ii)] $ \gamma=-p $ Substituting this value in Eq. (iv) $ \alpha \beta(-p)=-r \Rightarrow \alpha \beta=\frac{r}{p} $ Substituting this value in Eq. (iii) $ \begin{aligned} \frac{r}{p}+r(\alpha+\beta) & =+q \\ \frac{r}{p} & =+q \\ r & =p q \end{aligned} \quad[\because \alpha+\beta=0] $

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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