I can't seem to figure out how to solve this for x without graphing it:
0=x^3-3x+1
Ideally we would start with the rational roots theorem but here the only possible zeroes are -1 and 1 neither of which are zeroes.
Graphically is the best way to go here. the cubic equation is rather long and involved but:
If interested see Solving Cubic Equations
I would agree it would be simpler to use Newton's Method