Question

Solve the following quadratic equation:

`(x - 1)/(x - 2) + (x - 3)/(x - 4) = 3 1/3, x ≠ 2, 4`

Answer 1

`(x - 1)/(x - 2) + (x - 3)/(x - 4) = 3 1/3, x ≠ 2, 4` 

⇒ `((x - 1)(x - 4) + (x - 2)(x - 3))/((x - 2)(x - 4)) = 10/3` 

⇒ `(x^2 - 5x + 4 + x^2 - 5x + 6)/(x^2 - 6x + 8) = 10/3`  

⇒ `(2x^2 - 10x + 10)/(x^2 - 6x + 8) = 10/3`  

⇒ `(x^2 - 5x + 5)/(x^2 - 6x + 8) = 5/3` 

⇒ 3x² – 15x + 15 = 5x² – 30x + 40

⇒ 2x² – 15x + 25 = 0 

⇒ 2x² – 10x – 5x + 25 = 0 

⇒ 2x(x – 5) – 5(x – 5) = 0 

⇒ (x – 5)(2x – 5) = 0 

⇒ x – 5 = 0 or 2x – 5 = 0 

⇒ `x = 5` or `x = 5/2` 

Hence, 5 and `5/2` are the roots of the given equation.