2019 AMC 8 Problems/Problem 20
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Problem 20
How many different real numbers satisfy the equation
Solution
We have that if and only if
. If
, then
, giving 2 solutions. If
, then
, giving 2 more solutions. All four of these solutions work, so the answer is
. Further, the equation is a quartic in
, so by the Fundamental Theorem of Algebra, there can be at most four real solutions.
Videos Explaining Solution
https://youtu.be/5BXh0JY4klM (Uses a difference of squares & factoring method, different from above solutions)
https://www.youtube.com/watch?v=44vrsk_CbF8&list=PLLCzevlMcsWNBsdpItBT4r7Pa8cZb6Viu&index=2 ~ MathEx
See Also
2019 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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 AJHSME/AMC 8 Problems and Solutions |
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