Question

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

• A. $x^2+\alpha x-\beta =0$
• B. $x^2-[(\alpha +\beta )+\alpha \beta ]x-\alpha \beta (\alpha +\beta )=0$
• C. $x^2+[(\alpha +\beta )+\alpha \beta ]x+\alpha \beta (\alpha +\beta )=0$
• D. $x^2+[(\alpha +\beta )+\alpha \beta ]x-\alpha \beta (\alpha +\beta )=0$

Answer 1

The correct option is C $x^2+[(\alpha +\beta )+\alpha \beta ]x+\alpha \beta (\alpha +\beta )=0$

Hece, $\alpha +\beta =-b$

$\alpha \beta =-c$

$\therefore b=-(\alpha +\beta )$

$\therefore c=-\alpha \beta$

$\therefore b+c=-(\alpha +\beta +\alpha \beta )$

$\therefore bc=(\alpha +\beta )\left(\alpha \beta \right)$

$\therefore bc={\alpha }^2\beta +{\beta }^2\alpha$

$\therefore$ Equation is :-

$x^2-(-(\alpha +\beta +\alpha \beta )\lambda +{\alpha }^2\beta +{\beta }^2\alpha =0$

$=x^2+(\alpha +\beta +\alpha \beta )x+{\alpha }^2\beta +{\beta }^2\alpha =0$

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