Question

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

• A. $2$
• B. $4$
• C. $6$
• D. None of these

Answer 1

Answer: (B)

$4$

${\alpha }^2+{\beta }^2=\alpha +\beta$ ...(1)
${\alpha }^2{\beta }^2=\alpha \beta$ ...(2)
$\Rightarrow \alpha \beta (\alpha \beta -1)=0$
$\Rightarrow \alpha =0$ or $\beta =0$ or $\alpha \beta =1$
For $\alpha =0$
Substituting this in (1)
${\beta }^2-\beta =0$
$\Rightarrow \beta =0,1$
Hence roots are $0,0$ and $0,1$
Similarly for $\beta =0$
We get roots are $0,0$ and $0,1$
And for $\alpha \beta =1$
From (1)
${(\alpha +\beta )}^2-2\alpha \beta =\alpha +\beta$
substitute $t=\alpha +\beta$
$t^2-t-2=0\Rightarrow (t-2)(t+1)=0$
$\Rightarrow \alpha +\beta =2,-1$
From $\alpha +\beta =2$ and $\alpha \beta =1$
We get root $1,1$
And from $\alpha +\beta =-1$ and $\alpha \beta =1$
We get root $\omega ,{\omega }^2$
Hence total 4 quadratic equations are possible