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Question
Solve the following equation by factorization
`(x - 3)/(x + 3) + (x + 3)/(x - 3) = 2(1)/(2)`
Solution
`(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.
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