How many solutions does cosx = x^ 2 have?

Like any other polynomial, you judge the number of solutions by the highest degree of the polynomial. This is a second degree polynomial so it has 2 solutions. They are x = + [sqrt (cos x) ] and x = - [sqrt (cos x) ]. If you graph those on a calculator the positive portion of the graph is above the x axis and the negative portion is below the x axis. It's really cool looking. You should graph it to see.