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The complete solution set of the inequality (2 + sqrt(3))^(x² - x) + (2 - sqrt(3))^(x² - x) >= 14 is (-inf, a] union [b, inf). Find |a| + |b|.

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

Correct answer: 4

Solution

Let t = (2+sqrt(3))^(x²-x). Since (2+sqrt(3))(2-sqrt(3))=1, the second term equals 1/t. So t + 1/t >= 14. Since 2+sqrt(3) > 1, t is increasing in u = x²-x. t+1/t >= 14 means t >= 7+4sqrt(3) = (2+sqrt(3))² or t <= (2-sqrt(3))² = (2+sqrt(3))^(-2). So u >= 2 or u <= -2, i.e., x²-x >= 2 or x²-x <= -2. Case 1: x²-x-2 >= 0 => (x-2)(x+1) >= 0 => x <= -1 or x >= 2. Case 2: x²-x+2 <= 0, discriminant = 1-8 = -7 < 0, no real solution. So solution: x in (-inf, -1] union [2, inf). a = -1, b = 2. |a|+|b| = 1+2 = 3. But options include 3 but also 4. Rechecking: |a|+|b| = |-1|+|2| = 1+2 = 3.

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