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

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(B) Let the roots of the quadratic equation be $\alpha$ and $\beta$. According to the properties of roots,$\alpha + \beta = p + 3$ and $\alpha \beta =

Vedclass · Accepted Answer

$2$

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(B) Let the roots of the quadratic equation be $\alpha$ and $\beta$. According to the properties of roots,$\alpha + \beta = p + 3$ and $\alpha \beta = 5p - 2$. The sum of the squares of the roots is given by $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta$. Substituting the values,we get $\alpha^2 + \beta^2 = (p + 3)^2 - 2(5p - 2)$. Expanding this,$\alpha^2 + \beta^2 = p^2 + 6p + 9 - 10p + 4 = p^2 - 4p + 13$. To find the minimum value,we complete the square: $p^2 - 4p + 13 = (p^2 - 4p + 4) + 9 = (p - 2)^2 + 9$. The expression $(p - 2)^2 + 9$ attains its minimum value when $(p - 2)^2 = 0$,which implies $p = 2$.

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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