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Find the smallest positive integer p such that the expression x² - 2px + 3p + 4 is negative for at least one real value of x.

1. 3
2. 4
3. 5
4. 6

Correct answer: 5

Solution

The quadratic f(x) = x² - 2px + 3p + 4 opens upward (leading coefficient = 1 > 0). It can be negative for some real x only if it has two distinct real roots, i.e., discriminant D > 0. D = (-2p)² - 4(1)(3p + 4) = 4p² - 12p - 16. D > 0 means 4p² - 12p - 16 > 0, i.e., p² - 3p - 4 > 0, i.e., (p - 4)(p + 1) > 0. This holds when p > 4 or p < -1. Since we want the smallest positive integer p, the answer is p = 5.

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