Question

The number of quadratic equation which are unchanged by squaring their roots is...........

Answer 1

Let $\alpha$ $\beta$ be the roots of ${ax}^2+bx+c=0$ such that
${\alpha }^2=\alpha ,{\beta }^2=\beta$
$\therefore$ Sum of roots ${\alpha }^2+{\beta }^2=\alpha +\beta$ ...(1)
and product of roots ${\alpha }^2{\beta }^2=\alpha \beta$ ...(2)
Now equation (2) $\Rightarrow \alpha \beta (\alpha \beta -1)=0$
$\Rightarrow \alpha$ or $\beta =0$ or $\alpha \beta =1$
Case I: $\alpha =0$
equation (1) $\Rightarrow {\beta }^2=\beta \Rightarrow \beta =0,1$
So, $\beta =0\Rightarrow$ the equation is $x^2=0.$
and $\beta =1\Rightarrow$ the equation is $x^2-x=0.$
Case II: $\beta =0$
This case also given the same equation as in case I
Case III: $\alpha \beta =1$
equation (1) $\Rightarrow {(\alpha +\beta )}^2-2\left(1\right)=\alpha +\beta .$
$\Rightarrow {(\alpha +\beta )}^2-(\alpha +\beta )-2=0.$
$\Rightarrow (\alpha +\beta )=-1,2$
So $(\alpha +\beta )=-1\Rightarrow$ the equation is $x^2+(-1)x+1=0.$
$\Rightarrow x^2+x+1=0.$
and $\alpha +\beta =2\Rightarrow$ the equation is $x^2-2x+1=0$
$\therefore$ There are in all $4$ equation