Question

If $\alpha$and $\beta$are the roots of the quadratic equation $x^2+x+1=0$, then the equation, whose roots are ${\alpha }^{19}$and ${\beta }^7$, is

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

Answer 1

The correct option is D

$x^2+x+1=0$

Explanation for correct option:

Step 1. The given quadratic equation :

Given that $\alpha$and $\beta$ are the roots of the quadratic equation $x^2+x+1=0$

$$\begin{gathered} x^2+x+1=0 \\ \Rightarrow x=\frac{-1\pm \sqrt{1-4}}{2} \\ \Rightarrow x=\frac{-1\pm i\sqrt{3}}{2} \\ \Rightarrow x=\frac{-1+i\sqrt{3}}{2},\frac{-1-i\sqrt{3}}{2} \\ \Rightarrow x={\omega }^2,\omega \end{gathered}$$

$\Rightarrow \alpha =\omega ;\beta ={\omega }^2$

Step 2. Find the equation whose roots ${\mathit{\alpha }}^{\mathbf{19}}$and ${\mathit{\beta }}^{\mathbf{7}}$:

$$\begin{gathered} {\alpha }^{19}={\omega }^{19};{\beta }^7={\left({\omega }^2\right)}^7 \\ \Rightarrow {\alpha }^{19}=\omega ;{\beta }^7={\omega }^{14}={\omega }^2\mathbf{(}\mathbf{\because }\mathbf{}{\mathit{\omega }}^{\mathbf{3}}\mathbf{=}\mathbf{1}\mathbf{)} \end{gathered}$$

Therefore the required equation is

$x^2+x+1=0$

Hence, the correct option is D.