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(C) Given that $\alpha$ and $\beta$ are the roots of $x^2 + bx - c = 0$.By the relation between roots and coefficients:$\alpha + \beta = -b$ and $\alp

Vedclass · Accepted Answer

$x^2 + [(\alpha + \beta) + \alpha \beta]x + \alpha \beta(\alpha + \beta) = 0$

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(C) Given that $\alpha$ and $\beta$ are the roots of $x^2 + bx - c = 0$.By the relation between roots and coefficients:$\alpha + \beta = -b$ and $\alpha \beta = -c$.This implies $b = -(\alpha + \beta)$ and $c = -\alpha \beta$.We need to find the equation whose roots are $b$ and $c$.Sum of the roots = $b + c = -(\alpha + \beta) - \alpha \beta = -[(\alpha + \beta) + \alpha \beta]$.Product of the roots = $bc = [ -(\alpha + \beta) ] 	imes [ -\alpha \beta ] = \alpha \beta(\alpha + \beta)$.The required quadratic equation is $x^2 - (	ext{sum of roots})x + (	ext{product of roots}) = 0$.Substituting the values:$x^2 - (-[(\alpha + \beta) + \alpha \beta])x + \alpha \beta(\alpha + \beta) = 0$.$x^2 + [(\alpha + \beta) + \alpha \beta]x + \alpha \beta(\alpha + \beta) = 0$.

If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2 + bx - c = 0$,then the equation whose roots are $b$ and $c$ is:

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