Question

If $\alpha$ are $\beta$ are the roots of $x^2+x+1=0$ then find the equation whose roots ${\alpha }^2$ and ${\beta }^2$

• A. $x^2+x+1=0$
• B. $x^2+2x+1=0$
• C. $x^2+x+2=0$
• D. $x^2+2x+2=0$

Answer 1

Answer: (A)

$x^2+x+1=0$

For the given equation, sum of roots $=\alpha +\beta =-\frac{1}{1}=-1$
Product of roots $=\alpha \times \beta =\frac{1}{1}=1$
Now, ${\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2-2(\alpha \times \beta )=(-1{)}^2-2\left(1\right)=1-2=-1$
And ${\alpha }^2\times {\beta }^2=(\alpha \times \beta {)}^2=1$
Equation whose roots are ${\alpha }^2$ and ${\beta }^2$ is $x^2-\left(Sumofroots\right)x+Productofroots=0$
$=>x^2-({\alpha }^2+{\beta }^2)x+{\alpha }^2\times {\beta }^2=0$
$=>x^2-(-1)x+1=0$
$=>x^2+x+1=0$