Let $\alpha = cis \dfrac{2\pi}{7}, \beta = \alpha + \alpha^{2} + \alpha^{4}$ and $\gamma = \alpha^{3} + \alpha^{5} + \alpha^{6}$. Then $\beta$ and $\gamma$ are the roots of the equation

alpha = cis(2pi/7). beta = alpha + alpha^2 + alpha^4 and gamma = alpha^3 + alpha^5 + alpha^6. beta + gamma = sum of all roots of z^7-1=0 excluding 1, which is -1. beta * gamma = (alpha+alpha^2+alpha^4)(alpha^3+alpha^5+alpha^6) = alpha^4+alpha^6+alpha^7+alpha^5+alpha^7+alpha^8+alpha^7+alpha^9+alpha^10 = alpha^4+alpha^6+1+alpha^5+1+alpha+1+alpha^2+alpha^3 = 3 + (alpha+alpha^2+alpha^3+alpha^4+alpha^5+alpha^6) = 3 - 1 = 2. Equation: z^2 - (beta+gamma)z + beta*gamma = 0 => z^2 - (-1)z + 2 = 0 => z^2 + z + 2 = 0.