The value of p for which the sum of the squar | vedclass

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(B) Let the roots of the quadratic equation $x^2 - (p + 3)x + (5p - 2) = 0$ be $\alpha$ and $\beta$.From the properties of quadratic equations,we have

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$2$

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(B) Let the roots of the quadratic equation $x^2 - (p + 3)x + (5p - 2) = 0$ be $\alpha$ and $\beta$.From the properties of quadratic equations,we have the sum of roots $\alpha + \beta = p + 3$ and the product of roots $\alpha \beta = 5p - 2$.We want to minimize the sum of the squares of the roots,which is $S = \alpha^2 + \beta^2$.Using the identity $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta$,we substitute the values:$S = (p + 3)^2 - 2(5p - 2)$$S = p^2 + 6p + 9 - 10p + 4$$S = p^2 - 4p + 13$To find the minimum value,we complete the square:$S = (p^2 - 4p + 4) + 9$$S = (p - 2)^2 + 9$Since $(p - 2)^2 \ge 0$,the minimum value of $S$ occurs when $p - 2 = 0$,which means $p = 2$.

The value of $p$ for which the sum of the squares of the roots of the equation $x^2 - (p + 3)x + (5p - 2) = 0$ assumes its least value is:

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