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

(→‎Video Solution)
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==Solution==
 
==Solution==
−
We can substitute <math>5</math>, <math>3</math>, and <math>6</math> into the functions' definitions:
+
We have
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
 
(5 \, \blacklozenge \, 3) \, \bigstar \, 6 &= \left(5^2-3^2\right) \, \bigstar \, 6 \\
 
(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 \, \bigstar \, 6 \\
 
&= (16-6)^2 \\
 
&= (16-6)^2 \\
 
&= \boxed{\textbf{(D) } 100}.
 
&= \boxed{\textbf{(D) } 100}.
 
\end{align*}</cmath>
 
\end{align*}</cmath>
−
<i>~pog</i> ~MathFun1000 (Minor Edits)
+
<i>~pog</i>
  
−
==Video Solution 1==
+
==Video Solution==
 
https://www.youtube.com/watch?v=Ij9pAy6tQSg&t=91
 
https://www.youtube.com/watch?v=Ij9pAy6tQSg&t=91
 +
~Interstigation
  
 
==Video Solution 2==
 
==Video Solution 2==
 
https://youtu.be/YYuEBGoEK1Y
 
https://youtu.be/YYuEBGoEK1Y
  
−
==Video Solution 1==
+
~savannahsolver
 +
 
 +
==Video Solution==
 
https://youtu.be/Q0R6dnIO95Y?t=53
 
https://youtu.be/Q0R6dnIO95Y?t=53
 +
 +
~STEMbreezy
  
 
==See Also==  
 
==See Also==  
 
{{AMC8 box|year=2022|num-b=1|num-a=3}}
 
{{AMC8 box|year=2022|num-b=1|num-a=3}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 01:25, 20 January 2023

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

We have \begin{align*} (5 \, \blacklozenge \, 3) \, \bigstar \, 6 &= \left(5^2-3^2\right) \, \bigstar \, 6 \\ &= 16 \, \bigstar \, 6 \\ &= (16-6)^2 \\ &= \boxed{\textbf{(D) } 100}. \end{align*} ~pog

Video Solution

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

Video Solution 2

https://youtu.be/YYuEBGoEK1Y

~savannahsolver

Video Solution

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

~STEMbreezy

See Also

2022 AMC 8 (Problems • Answer Key • Resources)
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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