Question

Solve the following equation by factorization

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

Answer 1

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

Put `(x - 3)/(x + 3) = a,`

then `(x + 3)/(x - 3) = (1)/a`

∴ `a + (1)/a = (5)/(2)`

2a² + 2 = 5a
⇒ 2a² - 5a + 2 = 0
⇒ 2a² - a - 4a + 2 = 0
⇒ a(2a - 1) -2(2a - 1) = 0
⇒ (2a - 1) (a - 2) = 0
Either 2a - 1 = 0,
then a = `(1)/(2)`
or
a - 2 = 0,
then a = 2

(a) When a = `(1)/(2)`, then

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

⇒ 2x - 6 = x + 3
⇒ 2x - x = 3 + 6
⇒ x = 9
(b) when a = 2, then

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

2x + 6 = x - 3
⇒ 2x - x = -3 - 6
⇒ x = -9
∴ x = 9, -9.