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Difference between revisions of "2019 AMC 8 Problems/Problem 20"

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==Problem 20==
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==Problem==
 
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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==Solution 1==
 
==Solution 1==
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We have that <math>(x^2-5)^2 = 16</math> if and only if <math>x^2-5 = \pm 4</math>. If <math>x^2-5 = 4</math>, then <math>x^2 = 9 \implies x = \pm 3</math>, giving 2 solutions. If <math>x^2-5 = -4</math>, then <math>x^2 = 1 \implies x = \pm 1</math>, giving 2 more solutions. All four of these solutions work, so the answer is <math>\boxed{\textbf{(D)} 4}</math>. Further, the equation is a [[quartic Equation|quartic]] in <math>x</math>, so by the [https://artofproblemsolving.com/wiki/index.php/Fundamental_Theorem_of_Algebra Fundamental Theorem of Algebra], there can be at most four real solutions.
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We have that <math>(x^2-5)^2 = 16</math> if and only if <math>x^2-5 = \pm 4</math>. If <math>x^2-5 = 4</math>, then <math>x^2 = 9 \implies x = \pm 3</math>, giving 2 solutions. If <math>x^2-5 = -4</math>, then <math>x^2 = 1 \implies x = \pm 1</math>, giving 2 more solutions. All four of these solutions work, so the answer is <math>\boxed{\textbf{(D) }4}</math>. Further, the equation is a [[quartic Equation|quartic]] in <math>x</math>, so by the [https://artofproblemsolving.com/wiki/index.php/Fundamental_Theorem_of_Algebra Fundamental Theorem of Algebra], there can be at most four real solutions.
  
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==Solution 2==
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We can expand <math>(x^2-5)^2</math> to get <math>x^4-10x^2+25</math>, so now our equation is <math>x^4-10x^2+25=16</math>. Subtracting <math>16</math> from both sides gives us <math>x^4-10x^2+9=0</math>. Now, we can factor the left hand side to get <math>(x^2-9)(x^2-1)=0</math>. If <math>x^2-9</math> and/or <math>x^2-1</math> equals <math>0</math>, then the whole left side will equal <math>0</math>. Since the solutions can be both positive and negative, we have <math>4</math> solutions: <math>-3,3,-1,1</math> (we can find these solutions by setting <math>x^2-9</math> and <math>x^2-1</math> equal to <math>0</math> and solving for <math>x</math>). So, the answer is <math>\boxed{\textbf{(D) }4}</math>.
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~UnstoppableGoddess
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==Solution 3==
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Subtract 16 from both sides and factor using difference of squares:
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<cmath>(x^2 - 5)^2 = 16 </cmath>
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<cmath>(x^2 - 5)^2 - 16 =0 </cmath>
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<cmath>(x^2 - 5)^2 - 4^2 = 0 </cmath> 
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<cmath>[(x^2 - 5)-4][(x^2 - 5) + 4] = 0</cmath>
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<cmath>(x^2 - 9)(x^2 - 1) =0 </cmath>
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<cmath>(x+3)(x-3)(x+1)(x-1) = 0 </cmath>
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Quite obviously, this equation has <math>\boxed{\textbf{(D) }4}</math> solutions.
  
  
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Note: <math>x^2</math> can be two possible things and still be the same, and in <math>(x^2+4)^2</math>, <math>x^2+4</math> can also be negative or positive, so <math>2 \cdot 2 = 4</math>. This is a very fast solution.
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~TaeKim
  
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==Solution 2==
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==Video Solution by Math-X (First understand the problem!!!)==
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We can expand <math>(x^2-5)^2</math> to get <math>x^4-10x^2+25</math>, so now our equation is <math>x^4-10x^2+25=16</math>. Subtracting <math>16</math> from both sides gives us <math>x^4-10x^2+9=0</math>. Now, we can factor the left hand side to get <math>(x^2-9)(x^2-1)=0</math>. If <math>x^2-9</math> and/or <math>x^2-1</math> equals <math>0</math>, then the whole left side will equal <math>0</math>. Since the solutions can be both positive and negative, we have <math>4</math> solutions: <math>-3,3,-1,1</math> (we can find these solutions by setting <math>x^2-9</math> and <math>x^2-1</math> equal to <math>0</math> and solving for <math>x</math>). So the answer is <math>\boxed{\textbf{(D)} 4}</math>.
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https://youtu.be/IgpayYB48C4?si=EHbnc8zZoQ15Gfv6&t=6050
  
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~UnstoppableGoddess
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~Math-X
  
 
==Solution 3==
 
==Solution 3==
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https://youtu.be/5BXh0JY4klM (Uses a difference of squares & factoring method, different from above solutions)
 
https://youtu.be/5BXh0JY4klM (Uses a difference of squares & factoring method, different from above solutions)
  
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==Solution 5 (Fast)==
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==Solution 5 (video of Solution 1)==
 
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
  
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==Video Solution==
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== Video Solution by Pi Academy ==
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https://youtu.be/V3HxkJhSn08
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https://youtu.be/Ds8Nzjj6pXs?si=QAwrO_bZHrTj6cba
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~ smartschoolboy9
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== Video Solution 2 (Gateway to Harder Questions)==
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https://www.youtube.com/watch?v=J-E4SGEi3QE&t=2s
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https://youtu.be/V3HxkJhSn08 -Happytwin
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Solution detailing how to solve the problem: https://youtu.be/x4cF3o3Fzj8
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https://youtu.be/dh9uf5_ZB5Q ~ Education, the Study of Everything
  
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-Happytwin
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https://youtu.be/Xm4ZGND9WoY ~ Hayabusa1
  
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== Video Solution ==
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https://www.youtube.com/watch?v=TpsuRedYOiM&t=250s ~ SpreadTheMathLove
  
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Solution detailing how to solve the problem: https://www.youtube.com/watch?v=eXJnG96Xqp4&list=PLbhMrFqoXXwmwbk2CWeYOYPRbGtmdPUhL&index=21
 
  
 
==See Also==
 
==See Also==

Latest revision as of 09:15, 9 November 2024

Problem

How many different real numbers $x$ satisfy the equation \[(x^{2}-5)^{2}=16?\]

$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8$

Solution 1

We have that $(x^2-5)^2 = 16$ if and only if $x^2-5 = \pm 4$. If $x^2-5 = 4$, then $x^2 = 9 \implies x = \pm 3$, giving 2 solutions. If $x^2-5 = -4$, then $x^2 = 1 \implies x = \pm 1$, giving 2 more solutions. All four of these solutions work, so the answer is $\boxed{\textbf{(D) }4}$. Further, the equation is a quartic in $x$, so by the Fundamental Theorem of Algebra, there can be at most four real solutions.

Solution 2

We can expand $(x^2-5)^2$ to get $x^4-10x^2+25$, so now our equation is $x^4-10x^2+25=16$. Subtracting $16$ from both sides gives us $x^4-10x^2+9=0$. Now, we can factor the left hand side to get $(x^2-9)(x^2-1)=0$. If $x^2-9$ and/or $x^2-1$ equals $0$, then the whole left side will equal $0$. Since the solutions can be both positive and negative, we have $4$ solutions: $-3,3,-1,1$ (we can find these solutions by setting $x^2-9$ and $x^2-1$ equal to $0$ and solving for $x$). So, the answer is $\boxed{\textbf{(D) }4}$.

~UnstoppableGoddess

Solution 3

Subtract 16 from both sides and factor using difference of squares:


\[(x^2 - 5)^2 = 16\] \[(x^2 - 5)^2 - 16 =0\] \[(x^2 - 5)^2 - 4^2 = 0\] \[[(x^2 - 5)-4][(x^2 - 5) + 4] = 0\] \[(x^2 - 9)(x^2 - 1) =0\] \[(x+3)(x-3)(x+1)(x-1) = 0\]


Quite obviously, this equation has $\boxed{\textbf{(D) }4}$ 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

https://youtu.be/0AY1klX3gBo

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

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