Dear Macrakis I was the one who gave that exact solution in the first place The golden ratio naturally appears in the 5th roots of unity if you plot the roots in the complex plane they form the vertices of a pentagon This then explains why you can express a lot of trigonometric angles pi n 5 in terms of the golden ratio For example cos 18 b 4 sqrt 2 5 1 4 2 sqrt When you solve a quintic equation naturally the fifth roots of unity are involved hence why the radical solution of the quintic x 5 5x 12 0 can be expressed in terms of the golden ratio I hope it is clearer now