Question

Solve for x: a/x-a+b/x-b=2; x not equal to a.b

Answer 1

Hi friend! Here is a detailed answer to your question. NOTE: Instead of (^2) I've used ("). 
Your question: a / x-b + b / x-a = 2 
a(x-a) + b(x-b) / (x-a)(x-b) = 2 
ax - a" + bx - b" = 2 (x-a) (x-b) 
ax - a" + bx - b" = (2x-2a) (x-b) 
ax - a" + bx - b" = 2x" - 2bx - 2ax + 2ab 
-2x" + 3bx + 3ax - a" - b" - 2ab = 0 
2x" - 3bx - 3ax + a" + b" +2ab = 0 
2x" - 3 (a+b) x + (a+b)" = 0 
2x" - 2(a+b)x - 1(a+b)x + (a+b)" = 0 
2x [x - (a+b)] - (a+b) [x - (a+b)] = 0 
(2x - a - b ) (x - a - b) = 0 

Thus, either - 
2x - a - b = 0 
Therefore x = a+b / 2 

or - 

x - a - b = 0 
Therefore x = a + b 

Enjoy....