Difference between revisions of "2019 AMC 8 Problems/Problem 20"
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How many different real numbers <math>x</math> satisfy the equation <cmath>(x^{2}-5)^{2}=16?</cmath> | How many different real numbers <math>x</math> satisfy the equation <cmath>(x^{2}-5)^{2}=16?</cmath> | ||
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~TaeKim | ~TaeKim | ||
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| + | ==Video Solution by Math-X (First understand the problem!!!)== | ||
| + | https://youtu.be/IgpayYB48C4?si=EHbnc8zZoQ15Gfv6&t=6050 | ||
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| + | ~Math-X | ||
==Solution 3== | ==Solution 3== | ||
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https://www.youtube.com/watch?v=44vrsk_CbF8&list=PLLCzevlMcsWNBsdpItBT4r7Pa8cZb6Viu&index=2 ~ MathEx | https://www.youtube.com/watch?v=44vrsk_CbF8&list=PLLCzevlMcsWNBsdpItBT4r7Pa8cZb6Viu&index=2 ~ MathEx | ||
| − | ==Video Solution | + | == Video Solution by Pi Academy == |
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| + | https://youtu.be/Ds8Nzjj6pXs?si=QAwrO_bZHrTj6cba | ||
| − | + | ~ smartschoolboy9 | |
| − | ==Video Solution== | + | == Video Solution 2 (Gateway to Harder Questions)== |
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| − | - | + | https://www.youtube.com/watch?v=J-E4SGEi3QE&t=2s |
| − | + | https://youtu.be/V3HxkJhSn08 -Happytwin | |
Solution detailing how to solve the problem: https://youtu.be/x4cF3o3Fzj8 | Solution detailing how to solve the problem: https://youtu.be/x4cF3o3Fzj8 | ||
| − | + | https://youtu.be/dh9uf5_ZB5Q ~ Education, the Study of Everything | |
| − | https://youtu.be/dh9uf5_ZB5Q | ||
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| − | ~Education, the Study of Everything | ||
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| − | + | https://youtu.be/Xm4ZGND9WoY ~ Hayabusa1 | |
| − | https:// | ||
| + | https://www.youtube.com/watch?v=TpsuRedYOiM&t=250s ~ SpreadTheMathLove | ||
Latest revision as of 09:15, 9 November 2024
Contents
Problem
How many different real numbers 
satisfy the equation ![\[(x^{2}-5)^{2}=16?\]](http://latex.artofproblemsolving.com/d/1/8/d188c4697f3f7cf1889796a037bd60b6dae3b29e.png)
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:
![\[(x^2 - 5)^2 = 16\]](http://latex.artofproblemsolving.com/b/e/1/be1b88efca88930d266988540af92ce898f4898d.png)
![\[(x^2 - 5)^2 - 16 =0\]](http://latex.artofproblemsolving.com/b/6/1/b617549377b984460a7e2964271fdd22895b900f.png)
![\[(x^2 - 5)^2 - 4^2 = 0\]](http://latex.artofproblemsolving.com/0/d/9/0d9c002f8f719803ef6a268ac7ff9c6e34ba71df.png)
![\[[(x^2 - 5)-4][(x^2 - 5) + 4] = 0\]](http://latex.artofproblemsolving.com/a/4/8/a48aa54688812c2cba181df8629fc5c92c6fb2ba.png)
![\[(x^2 - 9)(x^2 - 1) =0\]](http://latex.artofproblemsolving.com/d/6/2/d62ae7379617e3b86043089d0948c10356693f03.png)
![\[(x+3)(x-3)(x+1)(x-1) = 0\]](http://latex.artofproblemsolving.com/b/c/0/bc0dcc818d65cb1105c40f433dbee65cdedb5951.png)
Quite obviously, this equation has solutions.
~TaeKim
Video Solution by Math-X (First understand the problem!!!)
https://youtu.be/IgpayYB48C4?si=EHbnc8zZoQ15Gfv6&t=6050
~Math-X
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 by Pi Academy
https://youtu.be/Ds8Nzjj6pXs?si=QAwrO_bZHrTj6cba
~ smartschoolboy9
Video Solution 2 (Gateway to Harder Questions)
https://www.youtube.com/watch?v=J-E4SGEi3QE&t=2s
https://youtu.be/V3HxkJhSn08 -Happytwin
Solution detailing how to solve the problem: https://youtu.be/x4cF3o3Fzj8
https://youtu.be/dh9uf5_ZB5Q ~ Education, the Study of Everything
https://youtu.be/Xm4ZGND9WoY ~ Hayabusa1
https://www.youtube.com/watch?v=TpsuRedYOiM&t=250s ~ SpreadTheMathLove
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. 
