To solve this quadratic equation for cos(x), we can let y = cos(x) and rewrite the equation as:

2y^2 - y - 1 = 0

Now, we can use the quadratic formula to solve for y:

y = [-(-1) ± √((-1)^2 - 42(-1))]/(2*2)
y = [1 ± √(1 + 8)]/4
y = [1 ± √9]/4
y = (1 ± 3)/4

Therefore, the solutions for y (cos(x)) are:

y = (1 + 3)/4 OR y = (1 - 3)/4
y = 1 OR y = -0.5

So, cos(x) = 1 OR cos(x) = -0.5

Therefore, the solutions for x are:
x = cos^(-1)(1) OR x = cos^(-1)(-0.5)

x = 0 OR x = 2π/3 or 4π/3

Hence, the solutions for the equation 2cos^2(x) - cos(x) - 1 = 0 are x = 0, x = 2π/3, and x = 4π/3.