2019 AMC 8 Problems/Problem 20
Contents
Problem 20
How many different real numbers satisfy the equation
Solution 1
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.
Solution 2
We can expand to get , so now our equation is . Subtracting from both sides gives us . Now, we can factor the left hand side to get . If and/or equals , then the whole left side will equal . Since the solutions can be both positive and negative, we have solutions: (we can find these solutions by setting and equal to and solving for ). So, the answer is .
~UnstoppableGoddess
Solution 3
Subtract 16 from both sides and factor using difference of squares:
Quite obviously, this equation has solutions.
~TaeKim
Solution 3
Associated Video - https://www.youtube.com/watch?v=Q5yfodutpsw
Solution 4
https://youtu.be/5BXh0JY4klM (Uses a difference of squares & factoring method, different from above solutions)
Solution 5 (video of Solution 1)
https://www.youtube.com/watch?v=44vrsk_CbF8&list=PLLCzevlMcsWNBsdpItBT4r7Pa8cZb6Viu&index=2 ~ MathEx
Video Solution
-Happytwin
Video Solution
Solution detailing how to solve the problem: https://youtu.be/x4cF3o3Fzj8
See Also
2019 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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