2022 AMC 10B Problems/Problem 7
- The following problem is from both the 2022 AMC 10B #7 and 2022 AMC 12B #4, so both problems redirect to this page.
Contents
Problem
For how many values of the constant 
will the polynomial 
have two distinct integer roots?
Solution 1
Let 
and 
be the roots of 
By Vieta's Formulas, we have 
and 
This shows that and must be distinct factors of 
The possibilities of 
are ![\[\pm\{1,36\},\pm\{2,18\},\pm\{3,12\},\pm\{4,9\}.\]](http://latex.artofproblemsolving.com/9/a/9/9a995e75c8baf10fff9deffb17fa9f4cdd457393.png)
Each unordered pair gives a unique value of 
Therefore, there are 
values of 
namely 
~stevens0209 ~MRENTHUSIASM ~
Solution 2
Note that must be an integer. By the quadratic formula, 
Since 
is a multiple of
, and 
have the same parity, so 
is an integer if and only if is a perfect square.
Let 
Then, 
Since is an integer and is even, 
and 
must both be even. Assuming that is positive, we get 
possible values of , namely 
, which will give distinct positive values of , but 
gives 
and 
, giving 
identical integer roots. Therefore, there are distinct positive values of Multiplying that by to take the negative values into account, we get 
values of
pianoboy
See Also
| 2022 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
| 2022 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 3 |
Followed by Problem 5 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
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