Difference between revisions of "2015 AMC 10B Problems/Problem 14"
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Let <math>a</math>, <math>b</math>, and <math>c</math> be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation <math>(x-a)(x-b)+(x-b)(x-c)=0</math>? | Let <math>a</math>, <math>b</math>, and <math>c</math> be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation <math>(x-a)(x-b)+(x-b)(x-c)=0</math>? | ||
| − | <math>\textbf{(A)} 15\qquad \textbf{(B)} 15.5\qquad \textbf{(C)} 16\qquad \textbf{(D)} 16.5\qquad \textbf{(E)} 17</math> | + | <math>\textbf{(A) }15\qquad \textbf{(B) }15.5\qquad \textbf{(C) }16\qquad \textbf{(D) }16.5\qquad \textbf{(E) }17</math> |
==Solution== | ==Solution== | ||
Revision as of 13:57, 6 March 2015
Problem
Let 
, 
, and 
be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation 
?
Solution
Expanding the equation and combining like terms results in 
. By Vieta's formulae the sum of the roots is ![$\dfrac{-[-(a+2b+c)]}{2}=\dfrac{a+2b+c}{2}$](http://latex.artofproblemsolving.com/3/7/4/3743e38f1afcc5182d2a9da5e198ff8df18800ec.png)
. To maximize this expression we want to be the largest, and from there we can assign the next highest values to and . So let 
, 
, and 
. Then the answer is 
.
See Also
| 2015 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 13 |
Followed by Problem 15 | |
| 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. 
