Difference between revisions of "2002 AMC 12A Problems/Problem 12"
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Revision as of 09:12, 4 July 2013
- The following problem is from both the 2002 AMC 12A #12 and 2002 AMC 10A #14, so both problems redirect to this page.
Problem
Both roots of the quadratic equation 
are prime numbers. The number of possible values of 
is
Solution
Consider a general quadratic with the coefficient of 
being 
and the roots being 
and 
. It can be factored as 
which is just 
. Thus, the sum of the roots is the negative of the coefficient of 
and the product is the constant term. (In general, this leads to Vieta's Formulas).
We now have that the sum of the two roots is 
while the product is . Since both roots are primes, one must be 
, otherwise the sum would be even. That means the other root is 
and the product must be 
. Hence, our answer is 
.
See Also
| 2002 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 11 |
Followed by Problem 13 |
| 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 | |
| 2002 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 13 |
Followed by Problem 15 | |
| 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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
