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Solve the equation x⁴ - 2x³ + 3x² - 2x + 1 = 0.
- x = 1
- x = -1
- x = (1 +- sqrt5)/2
- x = (3 +- sqrt5)/2
Correct answer: x = (1 +- sqrt5)/2
Solution
The coefficients are palindromic, so divide by x² and substitute y = x + 1/x. This reduces to a quadratic in y. Solving gives y, then solving x + 1/x = y for each y produces the roots. The real roots are x = (1 +- sqrt5)/2 (the others are complex).
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