I was playing around with Desmos Graphing Calculator and found that the two graphs below intersect at (-1,0), (0,-1), (1.618, 1.618), and (-0.618, -0.618). The latter two points are the golden ratio and the negative of the golden ratio plus 1.
Graphs:
- $y = x^2 - 1$
- $x = y^2 - 1$
Why do the graphs intersect at these points?
You have $x = (x^2-1)^2 -1 \iff x+1 = (x^2-1)^2 \iff x^4 - 2x^2 - x = 0$ so $$x(x^3 - 2x - 1) = x(x+1)(x^2-x-1) =0$$ so the roots are $x=0,-1, \phi, -\phi^{-1}$ by using the quadratic formula on the last term.