2015 AMC 10B Problems/Problem 14
Problem
Let , , and be three distinct one-digit numbers. What is the maximum value of the sum of the roots of the equation ?
Solution
Solution 1
Expanding the equation and combining like terms results in . By Vieta's formula the sum of the roots is . 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 .
Solution 2
Factoring out from the equation yields . Therefore the roots are and . Because must be the larger root to maximize the sum of the roots, letting and be and respectively yields the sum .
Solution 3
- no math here*
There are 2 cases. Case 1 is that and . Lets test that 1st. If , the maximum value for and is . Then and The next highest values are and so and . Therefore, .
See Also
2015 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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All AMC 10 Problems and Solutions |
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