Mathematics - Quadratic Equation Question with Solution | TestHub

$x^2-x+1=0\Rightarrow x=\frac{1\pm \sqrt{1-4}}{2}$ $x=\frac{1\pm \sqrt{3}i}{2}$ $\alpha =\frac{1}{2}+i\frac{\sqrt{3}}{2},\beta =\frac{1}{2}-\frac{i\sqrt{3}}{2}$ $\alpha =\cos \frac{\pi }{3}+i\sin \frac{\pi }{3},\beta =\cos \frac{\pi }{3}-i\sin \frac{\pi }{3}$ ${\alpha }^{2009}+{\beta }^{2009}=2\cos 2009\left(\frac{\pi }{3}\right)$ $=2\cos \left[668\pi +\pi +\frac{2\pi }{3}\right]=2\cos \left(\pi +\frac{2\pi }{3}\right)$ $=-2\cos \frac{2\pi }{3}=-2\left(-\frac{1}{2}\right)=1$