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

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==Solution 1==
 
==Solution 1==
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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>
+
Note that <math>f(x) - x^2</math> has roots <math>2, 3</math>, and <math>4</math>. Therefore, we may write <math>f(x) = a(x-2)(x-3)(x-4) +x^2</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.  
 +
 
 +
In the case of line <math>AB</math>, for example, we may write <math>a(x-2)(x-3)(x-4)+x^2-5x+6 = a(x-2)(x-3)(x-r_1)</math> for some real number <math>r_1</math>. Dividing both sides by <math>(x-2)(x-3)</math> gives <math>a(x-4)+1 = a(x-r_1)</math> or <math>r_1 = \frac {4a-1}{a}</math>.
 +
 
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For line <math>AC</math>, we have <math>a(x-2)(x-3)(x-4)+x^2-6x+8 = a(x-2)(x-4)(x-r_2)</math> for some real number <math>r_2</math>, which gives <math>a(x-3)+1 = a(x-r_2)</math> or <math>r_2 = \frac {3a-1}{a}</math>.
 +
 
 +
Repeating the process with line <math>BC</math> gives <math>r_3 = \frac {2a-1}{a}</math>.
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Since <math>r_1 + r_2 + r_3 = 24</math>, we have <math> \frac {4a-1}{a} + \frac {3a-1}{a} + \frac {2a-1}{a} = 24</math> or <math> \frac {9a-3}{a} = 24</math>. Solving for <math>a</math> gives <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>
  
 
Solution by vedadehhc
 
Solution by vedadehhc

Revision as of 12:43, 3 July 2024

Problem

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 1

Note that $f(x) - x^2$ has roots $2, 3$, and $4$. Therefore, we may write $f(x) = a(x-2)(x-3)(x-4) +x^2$. 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.

In the case of line $AB$, for example, we may write $a(x-2)(x-3)(x-4)+x^2-5x+6 = a(x-2)(x-3)(x-r_1)$ for some real number $r_1$. Dividing both sides by $(x-2)(x-3)$ gives $a(x-4)+1 = a(x-r_1)$ or $r_1 = \frac {4a-1}{a}$.

For line $AC$, we have $a(x-2)(x-3)(x-4)+x^2-6x+8 = a(x-2)(x-4)(x-r_2)$ for some real number $r_2$, which gives $a(x-3)+1 = a(x-r_2)$ or $r_2 = \frac {3a-1}{a}$.

Repeating the process with line $BC$ gives $r_3 = \frac {2a-1}{a}$.

Since $r_1 + r_2 + r_3 = 24$, we have $\frac {4a-1}{a} + \frac {3a-1}{a} + \frac {2a-1}{a} = 24$ or $\frac {9a-3}{a} = 24$. Solving for $a$ gives $a = - \frac{1}{5}$.

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

$\boxed{\textbf{No need to find the equations for the lines, really.}}$ 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 $2$, $3$, and $x_1$ are the three roots of $f(x) - bx-c$. The values 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 \[\frac{9a-1}{a} = 2 + 3 + x_1= 3 + 4 + x_2 = 2 + 4 + x_3\] Adding all three equations up, we get \[3\left(\frac{9a-1}{a}\right) = 18 + x_1 + x_2 + x_3 = 18 + 24\] Solving this equation, we get $a = -\frac{1}{5}$. We finish as Solution 1 does. $\boxed{\textbf{(D)}\frac{24}{5}}$.

- Mathdummy

Cleaned up by SSding

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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