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If the sum of the roots of the equation $x^2+p x+q=0$ is three times their difference, then which one of the following is true
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Verified Answer
The correct answer is:
$2 p^2=9 q$
Let $\alpha, \beta$ are roots of $x^2+p x+q=0$
So $\alpha+\beta=-p$ and $\alpha \beta=q$
Given that $(\alpha+\beta)=3(\alpha-\beta)=-p \Rightarrow \alpha-\beta=\frac{-p}{3}$
Now $(\alpha-\beta)^2=(\alpha+\beta)^2-4 \alpha \beta$
$\Rightarrow \frac{p^2}{9}=p^2-4 q$ or $2 p^2=9 q$
So $\alpha+\beta=-p$ and $\alpha \beta=q$
Given that $(\alpha+\beta)=3(\alpha-\beta)=-p \Rightarrow \alpha-\beta=\frac{-p}{3}$
Now $(\alpha-\beta)^2=(\alpha+\beta)^2-4 \alpha \beta$
$\Rightarrow \frac{p^2}{9}=p^2-4 q$ or $2 p^2=9 q$
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