$x^2-x+1=0$
$\Rightarrow x=\frac{1\pm\sqrt{3}i}{2}=-\omega,-\omega^2$
(Where $\omega$ is the cube root of unity)$
Product of roots $\alpha \beta=1$
$\implies \beta=\alpha^{-1}$
$\alpha^{2018}+\alpha^{-2018}=\alpha^{2018}+\beta^{2018}$
$=(-\omega)^{2018}+(-\omega^2)^{2018}$
$=(\omega^3)^{672}.\omega^2+(\omega^3)^{2*672}.\omega^4$
$=\omega^2+\omega^4$
$=\omega^2+ \omega$
$=-1$ (option A)
($\color{blue}{1+\omega+\omega^2=0\; and \; \omega^3=1}$)
Cube root of unity--
$x^3=1$
$\Rightarrow x^3-1=0$
$\Rightarrow (x-1)(x^2+x+1)=0$
$\Rightarrow x=1,\frac{-1\mp\sqrt 3 i}{2}$