2022 AMC 8 Problems/Problem 2

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

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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