2019 AMC 10A Problems/Problem 19
Problem
What is the least possible value of where is a real number?
Solution
Grouping the first and last terms and two middle terms gives which can be simplified as . Since squares are nonnegative, the answer is
Solution 2
Let . Then becomes
We can use difference of squares to get , and expand this to get .
Refactor this by completing the square to get , which has a minimum value of . The answer is thus
-WannabeCharmander
Solution 3 (using calculus)
Similar to Solution 1, grouping the first and last terms and the middle terms, we get .
Letting , we get the expression . Now, we can find the critical points of to minimize the function.
To minimize the result, we use . Hence, the minimum is , so
See Also
2019 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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