Let the two curves be defined by the equations and . The points of intersection of these two curves is defined as the solutions to the conditions and . Substituting the curve equations into the previous equation, we get , and since , this is equivalent to . Let for simplicity, and it follows that solving is equivalent to finding the x-coordinates of all the intersections between the two curves.
So for instance if you want to find the points of intersection of the equations:
You will want to solve , which simplifies to . The solutions are therefore , and substituting back these x-values into either of the curve equations will give you the y-coordinates of the points of intersection, which are respectively. Thus the points of intersection are and .