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Solve the equation x² + 3*|x| + 2 = 0 over the real numbers. What is its solution set?

1. No real solution (empty set)
2. x = -1, -2
3. x = 1, 2
4. x = -1, -2, 1, 2

Correct answer: No real solution (empty set)

Solution

Put t = |x|, t >= 0. Then t² + 3t + 2 = (t+1)(t+2) = 0 gives t = -1 or t = -2, both negative. Since |x| >= 0, neither is attainable. Also for real x, x² >= 0 and 3|x| >= 0, so x² + 3|x| + 2 >= 2 > 0 always. Hence no real solution.

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