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Solve the equation for real x: x^(2/3) + x^(1/3) - 2 = 0.

1. x = 1 and x = -8
2. x = 1 and x = 8
3. x = -1 and x = 8
4. x = 2 and x = -1

Correct answer: x = 1 and x = -8

Solution

Let t = x^(1/3). The equation becomes t² + t - 2 = 0, i.e. (t + 2)(t - 1) = 0, so t = 1 or t = -2. Then x = t³: t = 1 -> x = 1; t = -2 -> x = (-2)³ = -8. Both are real solutions.

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