Question

If $\alpha$ and $\beta$ are the roots of the equation $x^2+px+q=0$, then the equation whose roots are ${\alpha }^2+\alpha \beta$ and ${\beta }^2+\alpha \beta$ is

• A. $x^2+p^{2}x+p^{2}q=0$
• B. $x^2-q^{2}x+p^{2}q=0$
• C. $x^2+q^{2}x+p^{2}q=0$
• D. $x^2-p^{2}x+p^{2}q=0$

Answer 1

The correct option is D $x^2-p^{2}x+p^{2}q=0$

$x^2+px+q=0,$
$\therefore \alpha +\beta =-p;\alpha \beta =q$
$\therefore {\alpha }^2+\alpha \beta +{\beta }^2+\alpha \beta ={(\alpha +\beta )}^2=p^2$
$({\alpha }^2+\alpha \beta )({\beta }^2+\alpha \beta )=\alpha +\beta {(\alpha +\beta )}^2=q{p}^2$
$\therefore$ required equation is $x^2-p^{2}x+p^{2}q=0$