Question

The number of quadratic equations which are unchanged by squaring their roots are

• A. $2$
• B. $4$
• C. $6$
• D. $8$

Answer 1

Answer: (B)

$4$

Let $\alpha ,\beta$ be the roots of the quadratic equation.
$\alpha +\beta ={\alpha }^2+{\beta }^2$
$\alpha \beta ={\alpha }^2{\beta }^2$
Now, ${\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2-2\alpha \beta$
From equation, $\alpha \beta ={\alpha }^2{\beta }^2$, we can see that either $\alpha \beta =1$ or $0$.
When $\alpha \beta =1$, we get two solutions for $(\alpha ,\beta )$, they are $(\omega ,{\omega }^2)$ and $(1,1)$.
When $\alpha \beta =0$ roots are $(0,0)$ and $(1,0)$.
Hence, four solutions are possible.
Option B is correct.