Difference between revisions of "2020 AMC 10A Problems/Problem 5"
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==See Also== | ==See Also== |
Revision as of 15:08, 27 May 2020
Problem
What is the sum of all real numbers for which
Solution 1
Split the equation into two cases, where the value inside the absolute value is positive and nonpositive.
Case 1:
The equation yields , which is equal to . Therefore, the two values for the positive case is and .
Case 2:
Similarly, taking the nonpositive case for the value inside the absolute value notation yields . Factoring and simplifying gives , so the only value for this case is .
Summing all the values results in .
Solution 2
We have the equations and .
Notice that the second is a perfect square with a double root at , and the first has real roots. By Vieta's, the sum of the roots of the first equation is . .
Video Solution
~IceMatrix
https://www.youtube.com/watch?v=7-3sl1pSojc
~bobthefam
See Also
2020 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
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 |
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