SOLUTION: If A and B are positive integers such that both 4x2 +Ax+36 and 6x2+Bx+24 factor over the integers, find the largest possible value of A-B.

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Question 793182: If A and B are positive integers such that both 4x2 +Ax+36 and 6x2+Bx+24 factor over the integers, find the largest possible value of A-B.
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
If 4x%5E2%2BAx%2B36 can be factored, to find the factors I would start by looking for pairs of factors of 4%2A36=144
With A%3E0, A would be the sum of one of those factor pairs.
Here are the factor pairs and their sums:
1%2A144=144 1%2B144=145
2%2A72=144 2%2B72=74
3%2A48=144 3%2B48=51
4%2A36=144 4%2B36=40
6%2A24=144 6%2B24=30
8%2A18=144 8%2B18=26
12%2A12=144 12%2B12=24

If 6x%5E2%2BBx%2B24 can be factored, to find the factors I would start by looking for pairs of factors of 6%2A24=144
With B%3E0, B would be the sum of one of those factor pairs, which were already listed above.

To get the largest possible value of A-B, I need to pair possible value of A with the smallest possible value of B.
So the largest difference would be 145-24=highlight%28121%29

In that case the factorings would be
4x%5E2%2B145x%2B36=%284x%2B1%29%28x%2B36%29 and 6x%5E2%2B24x%2B24=6%28x%2B2%29%5E2