Difference between revisions of "2000 AMC 12 Problems/Problem 15"
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==Solution 1== | ==Solution 1== | ||
Let <math>y = \frac{x}{3}</math>; then <math>f(y) = (3y)^2 + 3y + 1 = 9y^2 + 3y+1</math>. Thus <math>f(3z)-7=81z^2+9z-6=3(9z-2)(3z+1)=0</math>, and <math>z = -\frac{1}{3}, \frac{2}{9}</math>. These sum up to <math>\boxed{\textbf{(B) }-\frac19}</math>. | Let <math>y = \frac{x}{3}</math>; then <math>f(y) = (3y)^2 + 3y + 1 = 9y^2 + 3y+1</math>. Thus <math>f(3z)-7=81z^2+9z-6=3(9z-2)(3z+1)=0</math>, and <math>z = -\frac{1}{3}, \frac{2}{9}</math>. These sum up to <math>\boxed{\textbf{(B) }-\frac19}</math>. | ||
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+ | ==Solution 2 (Similar) == | ||
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+ | This is quite trivially solved, as <math>3x = \dfrac{9x}{3}</math>, so <math>P(3x) = P(9x/3) = 81x^2 + 9x + 1 = 7</math>. <math>81x^2+9x-6 = 0</math> has solutions <math>-\frac{1}{3}</math> and <math>\frac{2}{9}</math>. Adding these yields a solution of <math>\boxed{\textbf{(B) }-\frac19}</math>. | ||
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+ | ~ icecreamrolls8 | ||
==Solution 2== | ==Solution 2== |
Revision as of 12:47, 2 November 2022
- The following problem is from both the 2000 AMC 12 #15 and 2000 AMC 10 #24, so both problems redirect to this page.
Contents
[hide]Problem
Let be a function for which . Find the sum of all values of for which .
Solution 1
Let ; then . Thus , and . These sum up to .
Solution 2 (Similar)
This is quite trivially solved, as , so . has solutions and . Adding these yields a solution of .
~ icecreamrolls8
Solution 2
Similar to Solution 1, we have The answer is the sum of the roots, which by Vieta's Formulas is .
~dolphin7
Solution 3
Set to get From either finding the roots (-3 and 2), or using Vieta's formulas, we find the sum of these roots to be Each root of this equation is times greater than a corresponding root of (because gives ), thus the sum of the roots in the equation is or .
Solution 4
Since we have , occurs at Thus, . We set this equal to 7:
. For any quadratic , the sum of the roots is . Thus, the sum of the roots of this equation is .
Note
All solutions that apply Vieta must check if the discriminant is zero, which in this case it isn't.
Video Solutions
https://m.youtube.com/watch?v=NyoLydoc3j8&feature=youtu.be
Video Solution 2
https://youtu.be/3dfbWzOfJAI?t=1300
~ pi_is_3.14
See also
2000 AMC 12 (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
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 |
2000 AMC 10 (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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.