2020 AMC 10A Problems/Problem 5
Problem 5
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
See Also
2020 AMC 10A (Problems • Answer Key • Resources) | ||
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Followed by Problem 6 | |
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