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Difference between revisions of "2017 AMC 12B Problems/Problem 23"

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First, we can define <math>f(x) = a(x-2)(x-3)(x-4) +x^2</math>, which contains points <math>A</math>, <math>B</math>, and <math>C</math>. Now we find that lines <math>AB</math>, <math>AC</math>, and <math>BC</math> are defined by the equations <math>y = 5x - 6</math>, <math>y= 6x-8</math>, and <math>y=7x-12</math> respectively. Since we want to find the <math>x</math>-coordinates of the intersections of these lines and <math>f(x)</math>, we set each of them to <math>f(x)</math>, and synthetically divide by the solutions we already know exist (eg. if we were looking at line <math>AB</math>, we would synthetically divide by the solutions <math>x=2</math> and <math>x=3</math>, because we already know <math>AB</math> intersects the graph at <math>A</math> and <math>B</math>, which have <math>x</math>-coordinates of <math>2</math> and <math>3</math>). After completing this process on all three lines, we get that the <math>x</math>-coordinates of <math>D</math>, <math>E</math>, and <math>F</math> are <math>\frac{4a-1}{a}</math>, <math>\frac{3a-1}{a}</math>, and <math>\frac{2a-1}{a}</math> respectively. Adding these together, we get <math>\frac{9a-3}{a} = 24</math> which gives us <math>a = -\frac{1}{5}</math>. Substituting this back into the original equation, we get <math>f(x) = -\frac{1}{5}(x-2)(x-3)(x-4) + x^2</math>, and <math>f(0) = -\frac{1}{5}(-2)(-3)(-4) + 0 = \boxed{\textbf{(D)}\frac{24}{5}}</math>
 
First, we can define <math>f(x) = a(x-2)(x-3)(x-4) +x^2</math>, which contains points <math>A</math>, <math>B</math>, and <math>C</math>. Now we find that lines <math>AB</math>, <math>AC</math>, and <math>BC</math> are defined by the equations <math>y = 5x - 6</math>, <math>y= 6x-8</math>, and <math>y=7x-12</math> respectively. Since we want to find the <math>x</math>-coordinates of the intersections of these lines and <math>f(x)</math>, we set each of them to <math>f(x)</math>, and synthetically divide by the solutions we already know exist (eg. if we were looking at line <math>AB</math>, we would synthetically divide by the solutions <math>x=2</math> and <math>x=3</math>, because we already know <math>AB</math> intersects the graph at <math>A</math> and <math>B</math>, which have <math>x</math>-coordinates of <math>2</math> and <math>3</math>). After completing this process on all three lines, we get that the <math>x</math>-coordinates of <math>D</math>, <math>E</math>, and <math>F</math> are <math>\frac{4a-1}{a}</math>, <math>\frac{3a-1}{a}</math>, and <math>\frac{2a-1}{a}</math> respectively. Adding these together, we get <math>\frac{9a-3}{a} = 24</math> which gives us <math>a = -\frac{1}{5}</math>. Substituting this back into the original equation, we get <math>f(x) = -\frac{1}{5}(x-2)(x-3)(x-4) + x^2</math>, and <math>f(0) = -\frac{1}{5}(-2)(-3)(-4) + 0 = \boxed{\textbf{(D)}\frac{24}{5}}</math>
  
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Solution by gorefeebuddie
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Solution by vedadehhc
  
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Note: This is a really good AMC 12 problem. It is one of those problems that they have every year.
 
 
==Solution 2==
 
==Solution 2==
 
No need to find the equations for the lines, really. First of all, <math>f(x) = a(x-2)(x-3)(x-4) +x^2</math>. Let's say the line <math>AB</math> is <math>y=bx+c</math>, and <math>x_1</math> is the <math>x</math> coordinate of the third intersection, then <math>2</math>, <math>3</math>, <math>x_1</math> are the three roots of <math>f(x) - bx-c</math>. Apparently the value of <math>b</math> and <math>c</math> have no effect on the sum of the 3 roots, because the coefficient of the <math>x^2</math> term is always <math>9a-1</math>. So we have,  
 
No need to find the equations for the lines, really. First of all, <math>f(x) = a(x-2)(x-3)(x-4) +x^2</math>. Let's say the line <math>AB</math> is <math>y=bx+c</math>, and <math>x_1</math> is the <math>x</math> coordinate of the third intersection, then <math>2</math>, <math>3</math>, <math>x_1</math> are the three roots of <math>f(x) - bx-c</math>. Apparently the value of <math>b</math> and <math>c</math> have no effect on the sum of the 3 roots, because the coefficient of the <math>x^2</math> term is always <math>9a-1</math>. So we have,  

Revision as of 18:52, 1 December 2019

Problem 23

The graph of $y=f(x)$, where $f(x)$ is a polynomial of degree $3$, contains points $A(2,4)$, $B(3,9)$, and $C(4,16)$. Lines $AB$, $AC$, and $BC$ intersect the graph again at points $D$, $E$, and $F$, respectively, and the sum of the $x$-coordinates of $D$, $E$, and $F$ is 24. What is $f(0)$?

$\textbf{(A)}\quad {-2} \qquad \qquad \textbf{(B)}\quad 0 \qquad\qquad \textbf{(C)}\quad 2 \qquad\qquad \textbf{(D)}\quad \dfrac{24}5 \qquad\qquad\textbf{(E)}\quad 8$

Solution

First, we can define $f(x) = a(x-2)(x-3)(x-4) +x^2$, which contains points $A$, $B$, and $C$. Now we find that lines $AB$, $AC$, and $BC$ are defined by the equations $y = 5x - 6$, $y= 6x-8$, and $y=7x-12$ respectively. Since we want to find the $x$-coordinates of the intersections of these lines and $f(x)$, we set each of them to $f(x)$, and synthetically divide by the solutions we already know exist (eg. if we were looking at line $AB$, we would synthetically divide by the solutions $x=2$ and $x=3$, because we already know $AB$ intersects the graph at $A$ and $B$, which have $x$-coordinates of $2$ and $3$). After completing this process on all three lines, we get that the $x$-coordinates of $D$, $E$, and $F$ are $\frac{4a-1}{a}$, $\frac{3a-1}{a}$, and $\frac{2a-1}{a}$ respectively. Adding these together, we get $\frac{9a-3}{a} = 24$ which gives us $a = -\frac{1}{5}$. Substituting this back into the original equation, we get $f(x) = -\frac{1}{5}(x-2)(x-3)(x-4) + x^2$, and $f(0) = -\frac{1}{5}(-2)(-3)(-4) + 0 = \boxed{\textbf{(D)}\frac{24}{5}}$

Solution by vedadehhc

Solution 2

No need to find the equations for the lines, really. First of all, $f(x) = a(x-2)(x-3)(x-4) +x^2$. Let's say the line $AB$ is $y=bx+c$, and $x_1$ is the $x$ coordinate of the third intersection, then $2$, $3$, $x_1$ are the three roots of $f(x) - bx-c$. Apparently the value of $b$ and $c$ have no effect on the sum of the 3 roots, because the coefficient of the $x^2$ term is always $9a-1$. So we have, \[\frac{9a-1}{a} = 2+3 + x_1=3+4+x_2 = 2+4+x_3\] Add them up we have \[3\frac{9a-1}{a} = 18 + x_1+x_2+x_3 = 18 +24\] Solve it, we get $a = -\frac{1}{5}$. $\boxed{\textbf{(D)}\frac{24}{5}}$.

- Mathdummy

See Also

2017 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
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

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