Question

If alpha and beta are the roots of the equation x minus A into x minus b + c equal to zero then the roots of the equation x minus Alpha into x minus beta equal to c are

Answer 1

$\textbf{Given:}$

$\text{ \alpha and \beta are roots of (x-a)(x-b)+c=0 }$

$\textbf{To find:}$

$\text{The roots of the equation (x-\alpha)(x-\beta)=c }$

$\textbf{Solution:}$

$\text{Consider,}$

$(x-a)(x-b)+c=0$

$x^2-(a+b)x+(ab+c)=0$

$\text{Then,}$

$\alpha+\beta=\dfrac{a+b}{1}$

$\implies\bf\alpha+\beta=a+b$

$\alpha\beta=\dfrac{ab+c}{1}$

$\implies\bf\alpha\beta=ab+c$

$\text{Now,}$

$(x-\alpha)(x-\beta)=c$

$x^2-(\alpha+\beta)x+\alpha\beta=c$

$x^2-(a+b)x+ab+c=c$

$x^2-(a+b)x+ab=0$

$x(x-a)-b(x-a)=0$

$(x-a)(x-b)=0$

$\implies\bf\,x=a,\;b$

$\textbf{Answer:}$

$\textbf{The roots of (x-\alpha)(x-\beta)=c are \bf\,a and \bf\,b }$

$\textbf{Find more:}$

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find k if alpha +beta=1/2alpha beta 

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