To make the writing much easier, let

`R_1^2=X` `R_2^2=Y` `r_1^2=x` `r_2^2=y`

So the equation becomes:

`kC=sqrt(X-Y)+sqrt(x-y)-A`, and we now need to solve for x.

Add A to both sides; subtract `sqrt(X-Y)` from both sides:

`kC+A-sqrt(X-Y)=sqrt(x-y)`

and then square both sides:

`(kC+A-sqrt(X-Y))^2=x-y`

Re-arrange:

`x=y+(kC+A-sqrt(X-Y))^2`

We now re-express it in the original notation:

`r_1^2=r_2^2+(kC+A-sqrt(R_1^2-R_2^2))^2`

We have solved the equation for `r_1^2`, as required. There is no need to expand out the last bracket on the RHS.