Question

Se $\mathsf{a}$ e $\mathsf{b}$ são as raízes de $\mathsf{x^2 + 1 = x}$, então determine o valor de $\mathsf{a^3 + b^3}$.

Answer 1

$a^3+b^3=(a+b)(a^2-ab+b^2)=(a+b)(a^2+b^2-ab)\\ \\ a+b=1\\ ab=1\\ \\ x^2-x+1=0\\ \\ \Delta=-3\\ \\x=\frac{1\pm i\sqrt3}{2}\\ \\ a^2=(\frac{1+i\sqrt3}{2})^2=\frac{-1+i\sqrt3}{2}\\ \\ b^2=(\frac{1-i\sqrt3}{2})^2=\frac{-1-i\sqrt3}{2}\\ \\ 1\cdot(\frac{-1-i\sqrt3}{2}+\frac{-1+i\sqrt3}{2}-1)=-2$