Solve for x, a/(ax-1) + b/(bx-1) = a+b

a/ax-1 + b/bx-1 =
a+b

=> ( a- ax) / ax  + ( b-bx)/ bx = (a+
b)

We will multiply by
abx^2

==> (a- ax)(b-bx) = (a+b)
abx^2

==> (ab - 2abx + abx^2 = (a+b)
abx^2

==> ab - 2abx = (a+b)( abx^2) -
abx^2

We will facro abx^2 from the right
side.

==> ab -2abx = abx^2 ( a+b
-1)

We will factor ab from the left
side.

==> ab ( -2x +1) = abx^2 ( a+
b-1)

Now we will divide by
ab.

==> (-2x+1) = x^2 ( a+ b
-1)

==> (a+b-1) x^2 +2x -1 =
0

Now we have a quadratic equation, we will solve for x in
terms of a and b.

==> x1= ( -2 + sqrt(4 +4(a+b-1) /
2(a+b-1)

            = ( -2+ 2sqrt(a+b) /
2(a+b-1)

           = (-1+ sqrt(a+b) /
(a+b-1)

==>x1= (-1+ sqrt(a+b) /
(a+b-1)

==> x2= ( -1-
sqrt(a+b) / (a+b-1)