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(B) Given the roots of $x^2 - 5x + 16 = 0$ are $\alpha$ and $\beta$. From the properties of quadratic equations,$\alpha + \beta = 5$ and $\alpha\beta

Vedclass · Accepted Answer

$p = -1, q = -56$

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(B) Given the roots of $x^2 - 5x + 16 = 0$ are $\alpha$ and $\beta$. From the properties of quadratic equations,$\alpha + \beta = 5$ and $\alpha\beta = 16$. For the equation $x^2 + px + q = 0$,the sum of roots is $-p = (\alpha^2 + \beta^2) + \frac{\alpha\beta}{2}$. Using the identity $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$,we get: $-p = (5^2 - 2(16)) + \frac{16}{2} = (25 - 32) + 8 = -7 + 8 = 1$. Thus,$p = -1$. The product of roots is $q = (\alpha^2 + \beta^2) 	imes \frac{\alpha\beta}{2}$. $q = (25 - 32) 	imes \frac{16}{2} = (-7) 	imes 8 = -56$. Therefore,$p = -1$ and $q = -56$.

If $\alpha$ and $\beta$ are the roots of the equation $x^2 - 5x + 16 = 0$,and if $\alpha^2 + \beta^2$ and $\alpha\beta/2$ are the roots of the equation $x^2 + px + q = 0$,then:

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