Difference between revisions of "2022 AMC 8 Problems/Problem 2"

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&= (16-6)^2 \
 
&= (16-6)^2 \
 
&= \boxed{\textbf{(D) } 100}.
 
&= \boxed{\textbf{(D) } 100}.
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\end{align*}</cmath>
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<i>~pog</i> ~MathFun1000 (Minor Edits)
  
 
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\end{align*}<math></math>
\end{align*}</cmath>
 
 
<i>~pog</i> ~MathFun1000 (Minor Edits)
 
<i>~pog</i> ~MathFun1000 (Minor Edits)
  
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<cmath> =\boxed{\textbf{(D) } 100} </cmath>
 
<cmath> =\boxed{\textbf{(D) } 100} </cmath>
  
~megaboy6679
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~[[megaboy6679]]
  
 
==Video Solution 1 by Math-X (First understand the problem!!!)==
 
==Video Solution 1 by Math-X (First understand the problem!!!)==

Revision as of 23:01, 16 October 2024

Problem

Consider these two operations: \begin{align*} a \, \blacklozenge \, b &= a^2 - b^2\\ a \, \bigstar \, b &= (a - b)^2 \end{align*} What is the output of $(5 \, \blacklozenge \, 3) \, \bigstar \, 6?$

$\textbf{(A) } {-}20 \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 100 \qquad \textbf{(E) } 220$

Solution 1

We can substitute $5$, $3$, and $6$ into the functions' definitions: \begin{align*} (5 \, \blacklozenge \, 3) \, \bigstar \, 6 &= \left(5^2-3^2\right) \, \bigstar \, 6 \\ (5 \, \blacklozenge \, 3) \, \bigstar \, 6 &= \left(25-9\right) \, \bigstar \, 6 \\ &= 16 \, \bigstar \, 6 \\ &= (16-6)^2 \\ &= \boxed{\textbf{(D) } 100}. \end{align*} ~pog ~MathFun1000 (Minor Edits)

\end{align*}$$ (Error compiling LaTeX. Unknown error_msg) ~pog ~MathFun1000 (Minor Edits)

Solution 2

We can find a general solution to any $((a \, \blacklozenge \, b) \, \bigstar \, c)$. \[((a \, \blacklozenge \, b) \, \bigstar \, c)\] \[=((a^2-b^2) \, \bigstar \, c)\] \[=(a^2-b^2-c)^2\] \[=a^4+b^4-(a^2)(b^2)-2(a^2)(c)-(b^2)(a^2)+2(b^2)(c)+(c^2)\] \[=5^4+3^4-(5^2)(3^2)-2(5^2)(6)-(3^2)(5^2)+2(3^2)(6)+(6^2)\] \[=625+81-225-300-225+108+36\] \[=\boxed{\textbf{(D) } 100}\]

~megaboy6679

Video Solution 1 by Math-X (First understand the problem!!!)

https://youtu.be/oUEa7AjMF2A?si=JlQdlwkdbIYFNJXj&t=123

~Math-X

Video Solution 2 (HOW TO THINK CREATIVELY!!!)

https://youtu.be/ytDV0GNc9Mw

~Education, the Study of Everything

Video Solution 3

https://www.youtube.com/watch?v=Ij9pAy6tQSg&t=91 ~Interstigation

Video Solution 4

https://youtu.be/YYuEBGoEK1Y

~savannahsolver

Video Solution 5

https://youtu.be/Q0R6dnIO95Y?t=53

~STEMbreezy

Video Solution 6

https://www.youtube.com/watch?v=c123kPqd11I

~harungurcan

See Also

2022 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
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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