Question

The number of quadratic equation(s), with real roots which remain unchanged even after squaring its roots, is

• A. $1$
• B. $0$
• C. $3$
• D. $2$

Answer 1

The correct option is C $3$

Let $a{x}^2+bx+c=0\cdots \left(1\right)$ has roots $\alpha ,\beta$
Equation having roots ${\alpha }^2,{\beta }^2$ is $f\left(\sqrt{x}\right)=0$
i.e., $ax+b\sqrt{x}+c=0$
$\Rightarrow ax+c=-b\sqrt{x}$
$\Rightarrow a^2{x}^2+(2ac-b^2)x+c^2=0\cdots \left(2\right)$

$\left(1\right)$ and $\left(2\right)$ represent the same equation.
$\Rightarrow a=a^2,b=2ac-b^2,c=c^2$
$\Rightarrow a=1$ as $a=0$ is not possible
and $c=0,1$

Case $1:$ $b=2ac-b^2$ and $a=1,c=0$
$\Rightarrow b+b^2=0$
$\Rightarrow b=0$ or $b=-1$
Possible quadratic equations are
$x^2=0$
$x^2-x=0$

Case $2:$ $b=2ac-b^2$ and $a=1,c=1$
$\Rightarrow b^2+b-2=0$
$\Rightarrow b=-2$ or $b=1$
Possible quadratic equations are
$x^2-2x+1=0$
$x^2+x+1=0,$ Not possible as it has non-real roots.