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**Punch** Given the curve $\displaystyle y=1-2cos^2x+cosx$ for $\displaystyle 0<x<2\pi$. Calculate the values of x for which the curve meets the x-axis.

I don't know where I went wrong but this is my attempt, textbook answer is $\displaystyle x=\frac{4\pi}{3}$ and $\displaystyle \frac{2\pi}{3}$

__Attempt__

Meet x-axis,$\displaystyle y=0$

$\displaystyle 1-2cos^2x+cosx=0$

$\displaystyle (2cosx+1)(-cosx+1)=0$

$\displaystyle cosx=\frac{-1}{2}[Q_2,Q_3] , cosx=1[Q_1,Q_4]$ your work is fine to here

$\displaystyle \alpha=2.0943rad$ & $\displaystyle \alpha=0$ these are 2 of 4 solutions which include 0 and 360 degrees.

$\displaystyle x=\pi-2.0943, x=\pi+2.0943, x=0(rej), x=2\pi(rej)$

=$\displaystyle 1.05rad = 5.23rad $