Question

Solve the following quadratic equation:

`(x - 4)/(x - 5) + (x - 6)/(x - 7) = 3 1/3, x ≠ 5, 7`

Answer 1

`(x - 4)/(x - 5) + (x - 6)/(x - 7) = 3 1/3, x ≠ 5, 7` 

⇒ `((x - 4)(x - 7) + (x - 5)(x - 6))/((x - 5)(x - 7)) = 10/3` 

⇒ `(x^2 - 11x + 28 + x^2 - 11x + 30)/(x^2 - 12x + 35) = 10/3` 

⇒ `(2x^2 - 22x + 58)/(x^2 - 12x + 35) = 10/3`

⇒ `(x^2 - 11x + 29)/(x^2 - 12x + 35) = 5/3` 

⇒ 3x² – 33x + 87 = 5x² – 60x + 175 

⇒ 2x² – 27x + 88 = 0 

⇒ 2x² – 16x – 11x + 88 = 0 

⇒ 2x(x – 8) – 11(x – 8) = 0 

⇒ (x – 8) (2x – 11) = 0 

⇒ x – 8 = 0 or 2x – 11 = 0 

⇒ `x = 8` or `x = 11/2` 

Hence, 8 and `11/2` are the roots of the given equation.