2010 AMC 12A Problems/Problem 21
Contents
Problem
The graph of lies above the line except at three values of , where the graph and the line intersect. What is the largest of these values?
Solutions
Solution 1
The values in which intersect at are the same as the zeros of .
Since there are zeros and the function is never negative, all zeros must be double roots because the function's degree is .
Suppose we let , , and be the roots of this function, and let be the cubic polynomial with roots , , and .
In order to find we must first expand out the terms of .
[Quick note: Since we don't know , , and , we really don't even need the last 3 terms of the expansion.]
All that's left is to find the largest root of .
Solution 2
The values in which intersect at are the same as the zeros of . We also know that this graph has 3 places tangent to the x-axis, which means that each root has to have a multiplicity of 2. Let the function be .
Applying Vieta's formulas, we get or . Applying it again, we get, after simplification, .
Notice that squaring the first equation yields , which is similar to the second equation.
Subtracting this from the second equation, we get . Now that we have the term, we can manpulate the equations to yield the sum of squares. or . We finally reach .
Since the answer choices are integers, we can guess and check squares to get in some order. We can check that this works by adding then and seeing . We just need to take the lowest value in the set, square root it, and subtract the resulting value from 5 to get .
Note: One could also multiply by 2 and subtract from to obtain The ordered triple {16,4,1} sums to 21, and the answer choices are all positive integers, therefore the answer is 4.
Alternative method:
After reaching and , we can algebraically derive .
Applying Vieta's formulas on the term yields .
Notice that , so
Subtracting this from yields , so , which means that , , and are the roots of the cubic , and it is not hard to find that these roots are , , and . The largest of these values is .
Solution 3
First, has exactly roots. Therefore, .
So,
By matching the coefficients of the first terms, we have
Solving the equations above, we have sets of solutions; first set of which is . Second set of which is . After squaring both sets, they are the same i.e. .
This is equal to . The largest root is
~Arcticturn
See also
2010 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
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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