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If x is a real number satisfying x + 1/x = 3, find all possible values of x² - 1/x².

1. 3*sqrt(5)
2. -5*sqrt(3)
3. -3*sqrt(5)
4. 5*sqrt(3)

Correct answer: 3*sqrt(5)

Solution

x² - 1/x² = (x-1/x)(x+1/x) = (+/-sqrt(5))(3) = +/-3*sqrt(5). Since x>0 and x+1/x=3, x could be either root giving +3*sqrt(5) or -3*sqrt(5) depending on which root is chosen.

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