for what value of c in the equation x^2 + c does the orbit of zero land on a cycle of period 4?

Question

this is discreet dynamics. I do know what values of c make the orbit of zero tend to a cycle of period 4, but i can't figure out when it is part of a 4-cycle.

i need to solve this equation for c so that x=0 is a solution

((((x^2 + c)^2 +c)^2 + c)^2 + c) - x = 0

I don't really know how to solve polynomials of degree 8

I'm kind of confused here. any help would help...

Answer 1

If x must be 0,
(((c^2 + c)^2 + c)^2 + c)^2 + c = 0
Rather than expand this and then attempt to factor or iterate a solution, iterate as it stands.
The only non-zero solution appears to be
c = - 1.985424253

Answer 2

Sorry, that's way above me. Good luck.