Difference between revisions of "2005 AMC 12B Problems/Problem 12"
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+ | {{duplicate|[[2005 AMC 12B Problems|2005 AMC 12B #12]] and [[2005 AMC 10B Problems|2005 AMC 10B #16]]}} | ||
== Problem == | == Problem == | ||
The [[quadratic equation]] <math>x^2+mx+n</math> has roots twice those of <math>x^2+px+m</math>, and none of <math>m,n,</math> and <math>p</math> is zero. What is the value of <math>n/p</math>? | The [[quadratic equation]] <math>x^2+mx+n</math> has roots twice those of <math>x^2+px+m</math>, and none of <math>m,n,</math> and <math>p</math> is zero. What is the value of <math>n/p</math>? | ||
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== See also == | == See also == | ||
+ | {{AMC10 box|year=2005|ab=B|num-b=15|num-a=17}} | ||
{{AMC12 box|year=2005|num-b=11|num-a=13|ab=B}} | {{AMC12 box|year=2005|num-b=11|num-a=13|ab=B}} | ||
[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] |
Revision as of 16:25, 12 July 2011
- The following problem is from both the 2005 AMC 12B #12 and 2005 AMC 10B #16, so both problems redirect to this page.
Problem
The quadratic equation has roots twice those of , and none of and is zero. What is the value of ?
Solution
Let have roots and . Then
so and . Also, has roots and , so
and and . Thus .
Indeed, consider the quadratics . See Vieta's formulas.
See also
2005 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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 |
2005 AMC 12B (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 |