I used half angle formula to get from sin (x/2) + cos x = 0 -->

2cos^2 x - cos x + 1 = 0, but is it possible to factor?

I graphed the original on my calculator and the answer appears to be x=pi, but I want to understand how to arrive at that answer. I also want to arrive at that answer using the tools I have covered in the class, which at this point means using the half angle formula and trig id's. I have worked out cos (x/2) - sin x = 0 with no problem as I can factor it after using 1/2 angle formula.