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2014 AMC 10B Problem 20

Problem 20 of 25HarderAlgebraCounting & Probability

For how many integers xx is the number x4−51x2+50x^4-51x^2+50 negative?

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Solution

First, note that x4−51x2+50x^4-51x^2+50 =(x2−50)(x2−1).= (x^2-50)(x^2-1). The product is negative exactly when its two factors have opposite signs. Since x2−50<x2−1x^2-50<x^2-1, this requires x2−50<0<x2−1x^2-50<0<x^2-1. Thus 1<x2<501<x^2<50, or 2≤∣x∣≤72\le |x|\le7. There are 66 positive and 66 negative integer solutions, for a total of 1212. Thus, the correct answer is C .

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Concepts: factoring · inequality · counting integers in a range

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.