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Difference between revisions of "2015 AMC 12B Problems/Problem 12"

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==Solution==
 
==Solution==
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Expand the polynomial. We get <math>(x-a)(x-b)+(x-b)(x-c)=x^2-(a+b)x+ab+x^2-(b+c)+bc=2x^2-(a+2b+c)x+(ab+bc).</math>
  
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Now, consider a general quadratic equation <math>ax^2+bx+c=0.</math> The two solutions to this are <cmath>\dfrac{-b}{2a}+\dfrac{\sqrt{b^2-4ac}}{2a}, \dfrac{-b}{2a}-\dfrac{\sqrt{b^2-4ac}}{2a}.</cmath> The sum of these roots is <cmath>\dfrac{-b}{a}.</cmath>
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Therefore, reconsidering the polynomial of the problem, the sum of the roots is <cmath>\dfrac{a+2b+c}{2}.</cmath> Now, to maximize this, it is clear that <math>b=9.</math> Also, we must have <math>a=8, b=7</math> (or vice versa). The reason <math>a,b</math> have to equal these values instead of larger values is because each of <math>a,b,c</math> is distinct.
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2015|ab=B|num-a=13|num-b=11}}
 
{{AMC12 box|year=2015|ab=B|num-a=13|num-b=11}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 22:58, 3 March 2015

Problem

Let $a$, $b$, and $c$ be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation $(x-a)(x-b)+(x-b)(x-c)=0$ ?

$\textbf{(A)}\; 15 \qquad\textbf{(B)}\; 15.5 \qquad\textbf{(C)}\; 16 \qquad\textbf{(D)} 16.5 \qquad\textbf{(E)}\; 17$

Solution

Expand the polynomial. We get $(x-a)(x-b)+(x-b)(x-c)=x^2-(a+b)x+ab+x^2-(b+c)+bc=2x^2-(a+2b+c)x+(ab+bc).$

Now, consider a general quadratic equation $ax^2+bx+c=0.$ The two solutions to this are \[\dfrac{-b}{2a}+\dfrac{\sqrt{b^2-4ac}}{2a}, \dfrac{-b}{2a}-\dfrac{\sqrt{b^2-4ac}}{2a}.\] The sum of these roots is \[\dfrac{-b}{a}.\]

Therefore, reconsidering the polynomial of the problem, the sum of the roots is \[\dfrac{a+2b+c}{2}.\] Now, to maximize this, it is clear that $b=9.$ Also, we must have $a=8, b=7$ (or vice versa). The reason $a,b$ have to equal these values instead of larger values is because each of $a,b,c$ is distinct.

See Also

2015 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

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