How to find intersections of two overlapping circles with complicated equations?

Option 1: Graph both equations to see where they intersect.  If they do not intersect at lattice points, then use the substitution method.

Option 2: Use substitution method of system of equations
Equation of a circle: (x-h)² + (y-k)² = r²
Choose one of the equations and solve for one of the variables
(I am choosing to solve for y)::
(y-k)² = r² - (x-h)²
|y - k| = √(r² - (x-h)²)
y - k = +/- √(r² - (x-h)²)
     y = k +/- √(r² - (x-h)²)

Now, substitute k +/- √(r² - (x-h)²) for y into the other equation to solve for x.
Substitute those x values into y = k +/- √(r² - (x-h)²) to solve for y.

Hope this makes sense!