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If x = sqrt((5 + 2*sqrt(6)) / (5 - 2*sqrt(6))), find the value of x² * (x - 10)².

1. 0
2. 1
3. 4
4. 25

Correct answer: 1

Solution

Multiply numerator and denominator of the fraction by (5 + 2*sqrt(6)): the denominator becomes (5)² - (2*sqrt(6))² = 25 - 24 = 1, so the fraction equals (5 + 2*sqrt(6))². Taking the square root gives x = 5 + 2*sqrt(6). Then x - 10 = 2*sqrt(6) - 5. The product x*(x-10) = (5 + 2*sqrt(6))*(2*sqrt(6) - 5) = 10*sqrt(6) - 25 + 24 - 10*sqrt(6) = -1. Therefore x²*(x-10)² = [x*(x-10)]² = (-1)² = 1.

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