Question

how can I find the points of intersection between y =x^2 and x = y^2

Answer 1

You have two equations and 2 unknowns. Take the y from the first equation (that is, take x^2) and put it in for the y in the second equation. This results in x = (x^2)^2 or x = x^4.
To solve this equation, put everything on one side of the equation
x - x^4 = 0
Now, factor this
x(1 - x^3) = 0
x (1 - x)(1 + x + x^2) = 0 (This assumes you know/remember how to factor the difference of two cubes)
setting each factor = to 0 gives x = 0, x = 1, and 2 non-Real answers (and I am assuming that you are only interested in the real answers). So, the answers are
(x = 0, y = 0) and (x = 1, y = 1)