Question

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

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

Answer 1

The correct option is B

$4$

We have to find a number of quadratic equation whose root on squaring their roots does not change

So, let $\alpha ,\beta$be the quadratic equation’s roots

Since the equation on squaring the roots should not change. So, all the relations for both the roots of the quadratic equation should not change.

Hence, the following equation can be formed as

$\alpha +\beta ={\alpha }^2+{\beta }^2$and another is${\alpha }^2,{\beta }^2$

${\alpha }^2{\beta }^2–\alpha \beta =0$

On solving the above equation taking common, we get,

$$\begin{gathered} \alpha \beta (\alpha \beta –1)=0 \\ \alpha \beta =0,1 \end{gathered}$$

Therefore, either $\alpha \beta =0$

So, possible ordered pair for$(\alpha ,\beta )=(0,0)$ and also $(\alpha ,\beta )=(1,0)$ and $(0,1)$

While for$\alpha \beta =1$

The possible ordered pairs are$(\alpha ,\beta )=(1,1)oreither(\alpha ,\beta )=(\omega ,{\omega }^2)as{\omega }^3=1$

Hence for all the other corresponding values clearly the above equation of relation of roots will not be satisfied and so there are a total four quadratic equations possible.

Hence, the correct option is (B)