Problem:
For how many integers n between 1 and 100 does x2+x−n factor into the product of two linear factors with integer coefficients?
Answer Choices:
A. 0
B. 1
C. 2
D. 9
E. 10
Solution:
Let x2+x−n=(x−a)(x+b) where a and b are consecutive positive integers with a<b. Then by completing this table
a:b:n=ab:122236………910901011110
we see that there are 9 values for n between 1 and 100.
OR
By the quadratic formula, x2+x−n=(x−x1)(x−x2) if and only if x1,x2=2−1±1+4n. Since every odd square is of the form 1+4n, the answer is the number of odd squares between 1+4⋅1 and 1+4⋅100. There are 9 odd squares in this range, the largest of which is 192.