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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,the sum of the roots is $\alph

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,the sum of the roots is $\alpha + \beta = -p$ and the product of the roots is $\alpha \beta = q$.According to the problem,the sum of the roots is three times their difference: $\alpha + \beta = 3(\alpha - \beta) = -p$.This implies $\alpha - \beta = -p/3$.We know the identity $(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha \beta$.Substituting the values,we get $(-p/3)^2 = (-p)^2 - 4q$.$p^2/9 = p^2 - 4q$.$4q = p^2 - p^2/9$.$4q = 8p^2/9$.$36q = 8p^2$,which simplifies to $2p^2 = 9q$.

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

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