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  4. The number of quadratic equations that are unchanged by squaring their roots is

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Q. The number of quadratic equations that are unchanged by squaring their roots is

NTA AbhyasNTA Abhyas 2020Complex Numbers and Quadratic Equations

Solution:

$\alpha +\beta =\alpha ^{2}+\beta ^{2} \, \& \, \alpha \beta =\alpha ^{2}\beta ^{2}\Rightarrow \alpha \beta =0$ or $1$ .
If $\alpha \beta = 0 ,$ then let, $\alpha = 0 \Rightarrow \beta = 0$ or $1$ .
If $\beta = \frac{1}{\alpha }$ ,
$\alpha + \frac{1}{\alpha } = \left(\alpha \right)^{2} + \frac{1}{\left(\alpha \right)^{2}} = \left(\alpha + \frac{1}{\alpha }\right)^{2} - 2$
$\Rightarrow \left(\alpha + \frac{1}{\alpha }\right)^{2} - \left(\alpha + \frac{1}{\alpha }\right) - 2 = 0 \Rightarrow \alpha + \frac{1}{\alpha } = 2$ or $- 1 \Rightarrow \alpha = 1$ or $\omega , \, \omega ^{2}$
Hence number of such equations are four $\left(0 , \, 0\right) , \, \left(0 , \, 1\right) , \, \left(1 , \, 1\right) \, \& \, \left(\omega , \, \left(\omega \right)^{2}\right)$