Question

Solve the following quadratic equation:

`a/((ax - 1)) + b/((bx - 1)) = (a + b), x ≠ 1/a, 1/b`

Answer 1

`a/((ax - 1)) + b/((bx - 1)) = (a + b) `

⇒ `[a/((ax - 1)) - b] + [b/((bx - 1)) - a] = 0`  

⇒ `(a - b(ax - 1))/(ax - 1) + (b - a(bx - 1))/(bx - 1) = 0` 

⇒ `(a - abx + b)/(ax - 1) - (a - abx + b)/(bx - 1) = 0` 

⇒ `(a - abx + b)[1/(ax - 1) + 1/((bx - 1))] = 0` 

⇒ `(a - abx + b)[((bx - 1) + (ax - 1))/((ax - 1)(bx - 1))] = 0` 

⇒ `(a - abx + b)[[(a + b)x - 2)/((ax - 1)(bx - 1))] = 0` 

⇒ (a – abx + b)[(a + b)x – 2] = 0 

⇒ a – abx + b = 0 or (a + b)x – 2 = 0 

⇒ `x = ((a + b))/(ab)` or `x = 2/((a + b))` 

Hence, the roots of the equation are `((a + b))/(ab)` and `2/((a + b))`.