what the equation says is:
x(current) = a*(x(previous))*(1 - 2x(previous)). apparently "a" here is some constant that represents some feature of the system.
so we start with t = 0, to get an initial value of x0. the next value x1, is:
x1 = ax0(1 - 2x0) = ax0 - 2a(x0)^2
the second value, x2, is:
x2 = ax1(1 - 2x1) = a(ax0 - 2a(x0)^2)(1 - 2(ax0 - 2a(x0)^2))
= (a^2x0 - 2a^2(x0)^2)(1 - 2ax0 + 4a(x0)^2) = a^2x0 - (2a^2 + 2a^3)(x0)^2 + 8a^3(x0)^3 - 8a^3(x0)^4
i am not sure what you mean by "solve" the equation. i can't think of some other formula for x_n solely in terms of x0.
i'm pretty sure that if one does exist, it's fairly complicated, and horrendous.
to calculate any given x_n, you just need to start with x0, and work your way to it.