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How many distinct real values of x satisfy the equation x + sqrt(x² + sqrt(x³ + 1)) = 1?

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

Correct answer: 1

Solution

Rearranging gives sqrt(x² + sqrt(x³+1)) = 1 - x. Squaring: x² + sqrt(x³+1) = (1-x)² = 1 - 2x + x², so sqrt(x³+1) = 1 - 2x. Squaring again: x³ + 1 = 1 - 4x + 4x², giving x³ - 4x² + 4x = 0, so x(x² - 4x + 4) = 0, i.e. x(x-2)² = 0. Solutions x = 0 or x = 2. Checking x = 2: need 1 - 2x = 1 - 4 = -3 >= 0, which fails. Only x = 0 is valid. Hence one solution.

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