Difference between revisions of "2022 AMC 10B Problems/Problem 7"
MRENTHUSIASM (talk | contribs) (→Solution 1) |
MRENTHUSIASM (talk | contribs) (→Solution 1) |
||
| Line 6: | Line 6: | ||
<math>\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 16</math> | <math>\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 16</math> | ||
| − | ==Solution | + | ==Solution== |
Let <math>p</math> and <math>q</math> be the roots of <math>x^{2}+kx+36.</math> By [[Vieta's Formulas]], we have <math>p+q=-k</math> and <math>pq=36.</math> | Let <math>p</math> and <math>q</math> be the roots of <math>x^{2}+kx+36.</math> By [[Vieta's Formulas]], we have <math>p+q=-k</math> and <math>pq=36.</math> | ||
Revision as of 17:28, 17 November 2022
- The following problem is from both the 2022 AMC 10B #7 and 2022 AMC 12B #4, so both problems redirect to this page.
Problem
For how many values of the constant 
will the polynomial 
have two distinct integer roots?
Solution
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 ~
See Also
| 2022 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 5 |
Followed by Problem 7 | |
| 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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
