If the absolute difference between the roots | vedclass

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(B) Let the roots of the equation $x^2 + px + 3 = 0$ be $\alpha$ and $\beta$.Given that $|\alpha - \beta| = \sqrt{p}$.Squaring both sides,we get $(\al

Vedclass · Accepted Answer

$4$

Vedclass · Answer

(B) Let the roots of the equation $x^2 + px + 3 = 0$ be $\alpha$ and $\beta$.Given that $|\alpha - \beta| = \sqrt{p}$.Squaring both sides,we get $(\alpha - \beta)^2 = p$.We know that $(\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta$.Substituting the coefficients from the quadratic equation,$\alpha + \beta = -p$ and $\alpha\beta = 3$.So,$(-p)^2 - 4(3) = p$.$p^2 - 12 = p \implies p^2 - p - 12 = 0$.Factoring the quadratic equation: $(p - 4)(p + 3) = 0$.Thus,$p = 4$ or $p = -3$.Since the difference between the roots is $\sqrt{p}$,$p$ must be non-negative for the roots to be real and the expression to be defined in the real number system. Therefore,$p = 4$.

If the absolute difference between the roots of the equation $x^2 + px + 3 = 0$ is $\sqrt{p}$,then $p = ......$

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