Equation Resolution – Do Solve[ ] and reduce[ ] functions have bugs in Mathematica?

contributions:

F[x_] : = x ^ 3 - 2 x + 1;
Solve[f[x] == 0, x]

Exit:

{{x -> 1}, {x -> 1/2 (-1 - Sqrt[5])}, {x -> 1/2 (-1 + Sqrt)[5])}}

But the 2nd item on the list is not a solution in fact:

NOT[F[1/2(1+Sqrt[f[1/2(1+Sqrt[F[1/2(-1+Sqrt[f[1/2(-1+Sqrt[5])]]

Exit:

-5,55112 * 10 ^ -17

Reducing to the same problem then how to force these two functions to return only exact solutions, since -5.55112 * 10 ^ -17 is only an approximation but is not a solution.

Can solve the problems of resolution and reduction, and what is the true meaning of this solution:

1/2 (-1 + Sqrt[5])?

Is there a way to ask Mathematica to provide a detailed explanation of how it came to this solution approximation (a sort of reverse tracing of the calculations)?