Question

Solve the following quadratic equations by factorization:

`1/((x-1)(x-2))+1/((x-2)(x-3))+1/((x-3)(x-4))=1/6`

Answer 1

We have been given,

`1/(x-1)(x-2)+1/(x-2)(x-3)+1/(x-3)(x-4)=1/6`

`((x-3)(x-4)+(x-1)(x-4)+(x-1)(x-2))/((x-1)(x-4)(x-2)(x-3))=1/6`

`(3(x^2-5x+6))/((x^2-5x+4)(x^2-5x+6))=1/6`

18 = x² - 5x + 4

x² - 5x - 14 = 0

x² - 7x + 2x - 14 = 0

x(x - 7) + 2(x - 7) = 0

(x + 2)(x - 7) = 0

Therefore,

x + 2 = 0

x = -2

or,

x - 7 = 0

x = 7

Hence, x = -2 or x = 7.