x + 1/x = 1

Can solve as quadratic:

x^2 - x + 1 = 0

x = (1 ± √3 i)/2

Or rearrange to get x^2 = x-1

and use to reduce the power.

then (x+1)^2 --> x^2 + 2x + 1 = 3x

So now find 3x + 1/(3x)

Choose x = (1 + √3 i)/2

Find 3(1 + √3 i)/2 + 2/3(1 + √3 i)

= 3(1 + √3 i)/2 + 2(1 - √3 i)/3(1 + √3 i)(1 - √3 i)

= 3(1 + √3 i)/2 + 2(1 - √3 i)/3(1+3)

= 3(1 + √3 i)/2 + (1 - √3 i)/6

= 5/3 + (4/3)√3 i

Although double checking on Wolfram Alpha gives a different result which, when I simplified it, worked out to the conjugate of my answer.  So either I made a math error or Wolfram Alpha did.