What you have done looks correct to me. The final equation is a quadratic in x, which you can solve to find the x-coordinates of the two points of intersection. I doubt whether there is a neater way of doing this.
I need to rearrange the following in order to find x.
I'm attempting to solve two circle equations simultaneously, I have subtracted one from the other and found y in terms of x, and substituted this back into one of the original equations in order to solve for x... but am not certain how to proceed!
I arrived at this point following this procedure:
Expanding the brackets ->
replacing (-a) with g and (-b) with f (apparently this is convention)->
collect together the non-x and non-y terms ->
gives the general circle equation (I believe!)
One representing each of the two intersecting circles ->
Subtract (2) from (1) ->
Using g, f and c as the results of the subtractions ->
Subtract c and gx and then divide by f ->
Substituting this back into equation (1) gives the equation which I need to solve. (Also at top of post).
I'm uncertain if I'm going about this in the best way, so if anybody has an easier alternative I would be very grateful.
I cannot guarantee the circles will be orthogonal so I cannot use kite geometry.
Any help/advice would be greatly appreciated.