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Solve for real x: x² + 1/x² - 5(x + 1/x) + 8 = 0. Which option gives the complete real solution set?

1. {1, (3 + sqrt5)/2, (3 - sqrt5)/2}
2. {1, 2, 3}
3. {(3 + sqrt5)/2, (3 - sqrt5)/2}
4. {1, -1, 2}

Correct answer: {1, (3 + sqrt5)/2, (3 - sqrt5)/2}

Solution

Substituting t = x + 1/x turns the equation into a quadratic in t; each valid t (|t| >= 2 for real x) then gives an equation x² - t x + 1 = 0.

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