Multiple choice

Let f(x) be a quadratic polynomial in x such that f(x) ≥ 0 for all real numbers x. If f(2) = 0 and f(4) = 6, then f(−2) is equal to

  1. 12

  2. 36

  3. 24

  4. 6
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Since f(x) >= 0 and f(2) = 0, (x - 2)^2 must be a factor. Let f(x) = a(x - 2)^2. Using f(4) = 6: a(4 - 2)^2 = 6, so 4a = 6, a = 1.5. Then f(-2) = 1.5 * (-2 - 2)^2 = 1.5 * 16 = 24.

AI explanation

Since the non-negative quadratic polynomial f(x) has a root at x = 2, its vertex is (2, 0) and its equation takes the form a(x - 2)^2. Using the given condition f(4) = 6, we substitute 4 for x to get a(4 - 2)^2 = 6, which yields 4a = 6 and a = 1.5. The polynomial is 1.5(x - 2)^2, so evaluating f(-2) gives 1.5(-2 - 2)^2 = 1.5(16) = 24.

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