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.