I'm struggling to rearrange to find x - simultaneous circle equations

Hi,

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!

$\displaystyle

x^2+((-gx-c)/f)^2+g_1x+f_1((-gx-c)/f)+c_1=0

$

I arrived at this point following this procedure:

$\displaystyle

(x-a)^2+(y-b)^2-r^2 = 0

$

Expanding the brackets ->

$\displaystyle

x^2+y^2-2ax-2by+a^2+b^2-r^2=0

$

replacing (-a) with g and (-b) with f (apparently this is convention)->

$\displaystyle

x^2+y^2+2gx+2fy+g^2+f^2-r^2=0

$

collect together the non-x and non-y terms ->

$\displaystyle

g^2+f^2-r^2=c

$

$\displaystyle

x^2+y^2+2gx+2fy+c=0

$

gives the general circle equation (I believe!)

One representing each of the two intersecting circles ->

$\displaystyle

(1) x^2+y^2+2g_1x+2f_1y+c_1=0

$

$\displaystyle

(2) x^2+y^2+2g_2x+2f_2y+c_2=0

$

Subtract (2) from (1) ->

$\displaystyle

(2g_1-2g_2)x+(2f_1-2f_2)y+c_1-c_2=0

$

Using g, f and c as the results of the subtractions ->

$\displaystyle

gx+fy+c=0

$

Subtract c and gx and then divide by f ->

$\displaystyle

y=(-gx-c)/f

$

Substituting this back into equation (1) gives the equation which I need to solve. (Also at top of post).

$\displaystyle

x^2+((-gx-c)/f)^2+g_1x+f_1((-gx-c)/f)+c_1=0

$

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.