2019 AMC 8 Problems/Problem 20
Contents
- 1 Problem 20
- 2 Solution 1
- 3 Solution 2
- 4 Solution 3
- 5 Solution 3
- 6 Solution 4
- 7 Solution 5 (video of Solution 1)
- 8 Video Solution 1 (Gateway to Harder Questions)
- 9 Video Solution
- 10 Video Solution
- 11 Video Solution
- 12 Video Solution by The Power of Logic(Problem 1 to 25 Full Solution)
- 13 Video Solution by SpreadTheMathLove
- 14 See Also
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 1 (Gateway to Harder Questions)
https://www.youtube.com/watch?v=J-E4SGEi3QE&t=2s
Video Solution
-Happytwin
Video Solution
Solution detailing how to solve the problem: https://youtu.be/x4cF3o3Fzj8
Video Solution
~Education, the Study of Everything
Video Solution by The Power of Logic(Problem 1 to 25 Full Solution)
~Hayabusa1
Video Solution by SpreadTheMathLove
https://www.youtube.com/watch?v=TpsuRedYOiM&t=250s
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.