Trig Problem, where am I going wrong?

Hello everyone, I'm new to these forums. The long story short is I am trying to finish high school via correspondence, and I need to do two math courses, each containing four units. I am on the fourth unit, the first three gave me very few problems. Anyways, here is the problem:

2cos^2x+cosx=0 Solve for the domain (0, 2pi) to nearest hundredth of a radian.

I did this:

Let k=cosx

2k^2+k

=k(2k+1) or

=cosx(2cosx+1)

cosx=0

x=cos^-1(0)

x=1.57 (this is pi/2 or 90degrees)

CAST RULE:

pi-1.57=4.71

pi+1.57=4.71

2pi-1.57=4.71

Therefor for the first zero, x=1.57, 4.71

For the second zero:

2cosx+1=0

2cosx=(-1)

cosx=(-1/2)

x=cos^-1(-1/2)

x=2.09 (this is 2pi/3 or 120degrees)

CAST RULE: since cosx is negative, the angle is in quadrants two and three:

pi-2.09=1.05

pi+2.09=5.23

So we have 1.57, 4.71, 1.05, 5.23

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However when I check my answer on mathway, it shows the solutions as:

1.57 (correct)

4.71 (correct)

2.09 (?)

4.19 (?)

I don't understand why my calculations for the second zero are wrong. If I use quadrants ONE and FOUR (where cosx should be positive), I get 2.09 and 4.19, which is what mathway shows as the correct answers.

Can someone tell me where I'm going wrong here? The second zero calculates to cosx=(-1/2), and cosx is negative in quadrants two and three, not one and four ... I don't see my error.