Question

Solve the following equation by factorization:

`(2x)/(x - 4) + (2x - 5)/(x - 3) = 25/3`

Answer 1

Given,

⇒ `(2x)/(x - 4) + (2x - 5)/(x - 3) = 25/3`

⇒ `(2x(x - 3) + (2x - 5)(x - 4))/((x - 4)(x - 3)) = 25/3`

⇒ `(2x^2 - 6x + 2x^2 - 8x - 5x + 20)/(x^2 - 3x - 4x + 12) = 25/3`

⇒ `(4x^2 - 19x + 20)/(x^2 - 7x + 12) = 25/3` 

⇒ 3(4x² – 19x + 20) = 25(x² – 7x + 12)

⇒ 12x² – 57x + 60 = 25x² – 175x + 300

⇒ 25x² – 175x + 300 – (12x² – 57x + 60) = 0

⇒ 25x² – 175x + 300 – 12x² + 57x – 60 = 0

⇒ 13x² – 118x + 240 = 0

⇒ 13x² – 78x – 40x + 240 = 0

⇒ 13x(x – 6) – 40(x – 6) = 0

⇒ (13x – 40)(x – 6) = 0

⇒ (13x – 40) or (x – 6) = 0   ...[Using zero-product rule] 

⇒ 13x = 40 or x = 6

⇒ x = `40/13` or x = 6

Hence, `x = {6, 40/13}`.