Question

A/(x-b)+b/(x-a)=2 factorise

Answer 1

The given equation is

\frac{a}{x-b} +\frac{b}{x-a} =2 .

Multiply both sides of the equation by (x-a)(x-b) .

(x-a)(x-b)(\frac{a}{x-b} +\frac{b}{x-a}) =2(x-a)(x-b)\\ a(x-a)+b(x-a) =2(x-a)(x-b)\\ a(x-a)+b(x-a) =2(x^2-(a+b)x+ab)\\ 2x^2-3(a+b)x+a^2+b^2+2ab=0\\ 2x^2-3(a+b)x+(a+b)^2=0\\ (2x-a-b)(x-a-b)=0\\ x=\frac{a+b}{2},a+b

Thus the solutions are x=\frac{a+b}{2},a+b