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(D) For the equation $x^2 - 5x + 16 = 0$,the sum of roots $\alpha + \beta = 5$ and the product of roots $\alpha\beta = 16$.We need to find the roots o

Vedclass · Accepted Answer

$p = -1, q = -56$

Vedclass · Answer

(D) For the equation $x^2 - 5x + 16 = 0$,the sum of roots $\alpha + \beta = 5$ and the product of roots $\alpha\beta = 16$.We need to find the roots of the second equation $x^2 + px + q = 0$.The first root is $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = (5)^2 - 2(16) = 25 - 32 = -7$.The second root is $\frac{\alpha\beta}{2} = \frac{16}{2} = 8$.The roots of $x^2 + px + q = 0$ are $-7$ and $8$.Sum of roots: $-7 + 8 = 1 = -p \implies p = -1$.Product of roots: $(-7)(8) = -56 = q \implies q = -56$.Thus,$p = -1$ and $q = -56$.

If $\alpha, \beta$ are the roots of the equation $x^2 - 5x + 16 = 0$ and $(\alpha^2 + \beta^2)$ and $\frac{\alpha\beta}{2}$ are the roots of the equation $x^2 + px + q = 0$,then:

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