Question

The number of quadratic equations (consider leading coefficient as $1$) with real roots which remain unchanged when their roots are squared, is

Answer 1

Let the two roots be $\alpha ,\beta$.
$f\left(x\right)=(x-\alpha )(x-\beta )=0$
Roots are squared ${\alpha }^2,{\beta }^2$
Roots remain same as even after squaring,
$\alpha ={\alpha }^2,\beta ={\beta }^2\Rightarrow \alpha =0,1\Rightarrow \beta =0,1$
(roots are real)

So, $\alpha =0,\beta =0\Rightarrow x^2=0$
$\alpha =1,\beta =1\Rightarrow (x-1{)}^2=0\alpha =0,\beta =1\Rightarrow x^2-x=0\alpha =1,\beta =0\Rightarrow x^2-x=0$

Hence, there are $3$ quadratic equations possible.