Question

Solve the following quadratic equation:
$\frac{a}{(x-b)} + \frac{b}{(x-a)} = 2$ , (x≠a, b)

Answer 1

⇒a/(x-b) + b/(x-a) = 2 
⇒a(x-a)+b(x-b)=2(x-a)(x-b)
⇒ax - a² + bx - b² = 2x² - 2bx - 2ax + 2ab
⇒2x² - 3ax - 3bx + a² + b² + 2ab = 0
⇒2x² - 3ax - 3bx + (a + b)² = 0
⇒x² -3/2(a+b)x +1/2(a+b)² = 0     Let a+b = m
so 
⇒x² - 3/2mx + m² = 0
⇒x² - 1/2 mx - mx + 1/2m² = 0
⇒x(x - 1/2m) - m(x - 1/2m) = 0
⇒(x - 1/2m)(x - m) = 0

So the solutions are

⇒x = 1/2 m = 1/2(a + b)
and 
⇒x = m  = a+b