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Difference between revisions of "2000 AMC 12 Problems/Problem 15"

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<cmath>\text {(A)}\ -1/3 \qquad \text {(B)}\ -1/9 \qquad \text {(C)}\ 0 \qquad \text {(D)}\ 5/9 \qquad \text {(E)}\ 5/3</cmath>
 
<cmath>\text {(A)}\ -1/3 \qquad \text {(B)}\ -1/9 \qquad \text {(C)}\ 0 \qquad \text {(D)}\ 5/9 \qquad \text {(E)}\ 5/3</cmath>
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== Video Solution ==
 
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https://youtu.be/3dfbWzOfJAI?t=1300
 
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~ pi_is_3.14
 
  
 
==Solution 1==
 
==Solution 1==
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<math>81z^2 + 9z +1 = 7 \Longrightarrow 81z^2 + 9z - 6 = 0</math>. For any quadratic <math>ax^2 + bx +c = 0</math>, the sum of the roots is <math>-\frac{b}{a}</math>. Thus, the sum of the roots of this equation is <math>-\frac{9}{81} = \boxed{\textbf{(B) }-\frac19}</math>.
 
<math>81z^2 + 9z +1 = 7 \Longrightarrow 81z^2 + 9z - 6 = 0</math>. For any quadratic <math>ax^2 + bx +c = 0</math>, the sum of the roots is <math>-\frac{b}{a}</math>. Thus, the sum of the roots of this equation is <math>-\frac{9}{81} = \boxed{\textbf{(B) }-\frac19}</math>.
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== Video Solution ==
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https://youtu.be/3dfbWzOfJAI?t=1300
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 +
~ pi_is_3.14
  
 
== See also ==
 
== See also ==

Revision as of 20:12, 1 May 2021

The following problem is from both the 2000 AMC 12 #15 and 2000 AMC 10 #24, so both problems redirect to this page.

Problem

Let $f$ be a function for which $f(\dfrac{x}{3}) = x^2 + x + 1$. Find the sum of all values of $z$ for which $f(3z) = 7$.

\[\text {(A)}\ -1/3 \qquad \text {(B)}\ -1/9 \qquad \text {(C)}\ 0 \qquad \text {(D)}\ 5/9 \qquad \text {(E)}\ 5/3\]

Solution 1

Let $y = \frac{x}{3}$; then $f(y) = (3y)^2 + 3y + 1 = 9y^2 + 3y+1$. Thus $f(3z)-7=81z^2+9z-6=3(9z-2)(3z+1)=0$, and $z = -\frac{1}{3}, \frac{2}{9}$. These sum up to $\boxed{\textbf{(B) }-\frac19}$.

Solution 2

Similar to Solution 1, we have $=81z^2+9z-6=0.$ The answer is the sum of the roots, which by Vieta's Formulas is $-\frac{b}{a}=-\frac{9}{81}=\boxed{\textbf{(B) }-\frac19}$.

~dolphin7

Solution 3

Set $f(\frac{x}{3}) = x^2+x+1=7$ to get $x^2+x-6=0.$ From either finding the roots or using Vieta's formulas, we find the sum of these roots to be $-1.$ Each root of this equation is $9$ times greater than a corresponding root of $f(3z) = 7$ (because $\frac{x}{3} = 3z$ gives $x = 9z$), thus the sum of the roots in the equation $f(3z)=7$ is $-\frac{1}{9}$ or $\boxed{\textbf{(B) }-\frac19}$.

Solution 4

Since we have $f(x/3)$, $f(3z)$ occurs at $x=9z.$ Thus, $f(9z/3) = f(3z) = (9z)^2 + 9z + 1$. We set this equal to 7:

$81z^2 + 9z +1 = 7 \Longrightarrow 81z^2 + 9z - 6 = 0$. For any quadratic $ax^2 + bx +c = 0$, the sum of the roots is $-\frac{b}{a}$. Thus, the sum of the roots of this equation is $-\frac{9}{81} = \boxed{\textbf{(B) }-\frac19}$.

Video Solution

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

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