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

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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) **

*a*/(

*ax*– 1) +

*b*/(

*bx*– 1) =

*a*+

*b*⇒

*a*–

*a*/(

*ax*– 1) +

*b*–

*b*/(

*bx*– 1) = 0 ⇒ [

*a*(

*ax*– 2)(

*bx*– 1) +

*b*(

*bx*– 2)(

*ax*– 1)] / [(

*ax*– 1) (

*bx*– 1)] = 0 ⇒

*ab*(

*a*+

*b*)

*x*– [(

*a*+

*b*)2 +2

*ab*]

*x*+ 2 (

*a*+

*b*) = 0 ⇒

*ab*(

*a*+

*b*)

*abx*– (

*a*+

*b*)2

*x*– 2

*abx*+ 2 (

*a*+

*b*) = 0 ⇒ (

*a*+

*b*)

*x*[

*abx*– (

*a*+

*b*)] – 2[

*abx*– (

*a*+

*b*)] = 0 ⇒ [(

*a*+

*b*)

*x*– 2][

*abx*– (

*a*+

*b*)] = 0 ⇒ [(

*a*+

*b*)

*x*– 2] = 0 or [

*abx*– (

*a*+

*b*)] = 0 ⇒

*x*= 2/(

*a*+

*b*) or

*x*= (

*a*+

*b*)/

*ab*