If alpha and beta are roots of x^{2}-x+1=0, t | vedclass

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(B) The given quadratic equation is $x^{2}-x+1=0$.The roots of this equation are given by $x = \frac{1 \pm \sqrt{1-4}}{2} = \frac{1 \pm i\sqrt{3}}{2}$

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-$2$

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(B) The given quadratic equation is $x^{2}-x+1=0$.The roots of this equation are given by $x = \frac{1 \pm \sqrt{1-4}}{2} = \frac{1 \pm i\sqrt{3}}{2}$.These roots are $-\omega$ and $-\omega^{2}$, where $\omega$ is the complex cube root of unity.Let $\alpha = -\omega$ and $\beta = -\omega^{2}$.Then, $\alpha^{2013} + \beta^{2013} = (-\omega)^{2013} + (-\omega^{2})^{2013}$.Since $2013$ is an odd number, $(-\omega)^{2013} = -\omega^{2013} = -(\omega^{3})^{671} = -(1)^{671} = -1$.Similarly, $(-\omega^{2})^{2013} = -(\omega^{2})^{2013} = -\omega^{4026} = -(\omega^{3})^{1342} = -(1)^{1342} = -1$.Therefore, $\alpha^{2013} + \beta^{2013} = -1 + (-1) = -2$.

If $\alpha$ and $\beta$ are roots of $x^{2}-x+1=0,$ then the value of $\alpha^{2013}+\beta^{2013}$ is

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