Your equation says that the y values on the two curves for a given x are equal -- so, yes, its solutions are the x-coordinates of the (two) intersections.
For each such solution, you can plug it into either of the original equations to find the corresponding y. You will get the same value of y from each equation (assuming you solved correctly), because that's what your equation says will happen. Did you try doing that? You won't get four points, just two.
The general formula is a little complex, so you have to be careful plugging into the quadratic, but it's worth the experience.