If the sum of the roots of the equation x^2 + | vedclass

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Question

(C) Let $\alpha$ and $\beta$ be the roots of the equation $x^2 + px + q = 0$.From the properties of quadratic equations,we have $\alpha + \beta = -p$

Vedclass · Accepted Answer

$2p^2 = 9q$

Vedclass · Answer

(C) Let $\alpha$ and $\beta$ be the roots of the equation $x^2 + px + q = 0$.From the properties of quadratic equations,we have $\alpha + \beta = -p$ and $\alpha\beta = q$.According to the problem,the sum of the roots is three times their difference: $(\alpha + \beta) = 3(\alpha - \beta)$.Substituting the sum,we get $-p = 3(\alpha - \beta)$,which implies $(\alpha - \beta) = -\frac{p}{3}$.We know the identity $(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$.Substituting the values,we get $(-\frac{p}{3})^2 = (-p)^2 - 4q$.$\frac{p^2}{9} = p^2 - 4q$.Multiplying by $9$,we get $p^2 = 9p^2 - 36q$.$36q = 8p^2$.Dividing by $4$,we get $9q = 2p^2$ or $2p^2 = 9q$.

If the sum of the roots of the equation $x^2 + px + q = 0$ is three times their difference,then which of the following is true?

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