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Question
Solve the following equation by factorization
`(x + 1)/(x - 1) + (x - 2)/(x + 2)` = 3
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Solution
`(x + 1)/(x - 1) + (x - 2)/(x + 2)` = 3
⇒ `((x + 1)(x + 2)+ (x - 2)(x -1))/((x - 1)(x + 2)` = 3
⇒ `(x^2 + 2x + x + 2 + x^2 - x - 2x + 2)/(x^2 + 2x - x - 2)`
⇒ `(x^2 + 3x + 2 + x^2 - 3x + 2)/(x^2 + x - 2) = (3)/(1)`
⇒ 2x2 + 4 = 3x2 + 3x - 6
⇒ 2x2 + 4 - 3x2 - 3x + 6 = 0
⇒ -x2 - 3x + 10 = 0
⇒ x2 + 3x - 10 = 0
⇒ x2 + 5x - 2x - 10 = 0
⇒ x(x + 5) - 2(x + 5) = 0
⇒ (x + 5) (x - 2) = 0
Either x + 5 = 0,
then x = -5
or
x - 2 = 0,
then x = 2
Hence x = -5, 2.
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