Question

Solve the following equation by using formula :

`(1)/(x - 2) + (1)/(x - 3) + (1)/(x - 4)` = 0

Answer 1

`(1)/(x - 2) + (1)/(x - 3) + (1)/(x - 4)` = 0

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

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

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

(2x - 5)(x - 4) = -1(x² - 5x + 6)
⇒ 2x² - 8x - 5x + 20 = -x² + 5x - 6
⇒ 2x² - 8x - 5x + 20 + x² - 5x + 6 = 0
⇒ 3x² - 18x + 26 = 0
Here, a = 3, b = -18, c = 26

∴ x = `(-b ± sqrt(b^2 - 4ac))/(2a)`

= `(-(-18) ± sqrt((-18)^2 - 4 xx 3 xx 26))/(2 xx 3)`

= `(18 ± sqrt(324 - 312))/(6)`

= `(18 ± sqrt(12))/(6)`

= `(18 ± 2sqrt(3))/(6)`

= `(9 ± sqrt(3))/(3)`   ...(Dividing by 2)

∴ x = `(9 + sqrt(3))/(3), (9 - sqrt(3))/(3)`

= `3 + sqrt(3)/(3), 3 - sqrt(3)/(3)`

= `3 + (1)/sqrt(3), 3 - (1)/sqrt(3)`.