2014 AMC 10B Problem 20

Question

For how many integers

x

is the number

x^4-51x^2+50

negative?

8

10

12

14

16

Factor the expression as a product in

x^2

Find where the two factors have opposite signs

First, note that

x^4-51x^2+50

= (x^2-50)(x^2-1).

The product is negative exactly when its two factors have opposite signs. Since

x^2-50<x^2-1

, this requires

x^2-50<0<x^2-1

. Thus

1<x^2<50

, or

2\le |x|\le7

. There are

6

positive and

6

negative integer solutions, for a total of

12

.

Thus, the correct answer is C .

Problem 20 in other years