Question

难题来了，大神们看图，求解求解





Answer 1

x+1/x +1 =0
x^2+x+1 =0
(x-1)(x^2+x+1) =0
x^3 -1 =0
x^3 =1
x^2019 +1/x^2019

=(x^3)^673 + 1/(x^3)^673
=1 + 1/1
=2

Answer 2

x+1/x+1=0
x²+x+1=0，△<0，这个方程的根为复数根。
x=-1/2±√3i/2
x=cos(2π/3)+isin(2π/3)或x=cos(4π/3)+isin(4π/3)
（1）x=cos(2π/3)+isin(2π/3)时，
x^2019+x^{-2019}
=cos(2π/3*2019)+isin(2π/3*2019)+cos(-2π/3*2019)+isin(-2π/3*2019)
=cos(673*2π)+isin(673*2π)+cos(-673*2π)-isin(673*2π)
=2cos(673*2π)
=2cos0
=2
（2）x=cos(4π/3)+isin(4π/3)时
x^2019+x^{-2019}
=cos(4π/3*2019)+isin(4π/3*2019)+cos(-4π/3*2019)+isin(-4π/3*2019)
=cos(673*4π)+isin(673*4π)+cos(-673*4π)-isin(673*4π)
=2cos(673*4π)
=2cos0
=2
综上，x^{2019}+1/x^{2019}=2