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

$\begin{array}{l}Let\alpha ,\beta arerootsofQuadraticEquation\\ and{\alpha }^2,{\beta }^{2}arerootsofanotherQuadraticEquation\\ \because TheQuadraticEquationremainsthesame\\ \therefore \alpha +\beta ={\alpha }^2+{\beta }^2--\left(i\right)and\alpha \beta ={\alpha }^2{\beta }^2--\left(ii\right)\\ Now\alpha \beta ={\alpha }^2{\beta }^2\Rightarrow \alpha \beta (\alpha \beta -1)=0\\ \Rightarrow \alpha =0or\beta =0or\alpha \beta =1\\ If\alpha =0thenfrom\left(1\right)\beta ={\beta }^2\\ \Rightarrow \beta (1-\beta )=0\\ \Rightarrow \beta =0,\beta =1\\ \therefore \alpha =0,\beta =0and\alpha =0,\beta =1aretwosetofvalues\\ If\alpha \beta =1then\alpha +\frac{1}{\alpha }={\alpha }^2+\frac{1}{{\alpha }^2}from\left(1\right)\\ \Rightarrow \alpha +\frac{1}{\alpha }={(\alpha +\frac{1}{\alpha })}^2-2\\ \Rightarrow {(\alpha +\frac{1}{\alpha })}^2-(\alpha +\frac{1}{\alpha })-2=0\\ \therefore \alpha +\frac{1}{\alpha }=2or\alpha +\frac{1}{\alpha }=-1\\ \Rightarrow {\alpha }^2-2\alpha +1=0or{\alpha }^2+\alpha +1=0\\ \Rightarrow \alpha =1,\Rightarrow \alpha =\omega ,{\omega }^2\\ It\alpha =1then\beta =1[\because \alpha \beta =1]\\ If\alpha =\omega then\beta ={\omega }^{2}andIf\alpha ={\omega }^{2}then\beta =\omega \\ \therefore \alpha =0,\beta =0and\alpha =0,\beta =1ad\alpha =1,\beta =1and\alpha =\omega ,\beta =\omega \\ \therefore 4setsofvaluesof\alpha ,\beta \end{array}$