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

Revision as of 14:26, 28 January 2022

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 value 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

2022 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 1	Followed by Problem 3
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All AJHSME/AMC 8 Problems and Solutions

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

Revision as of 14:26, 28 January 2022

Problem

Solution

See Also

@@ Line 13: / Line 13: @@
 We have
 <cmath>\begin{align*}
-(5 \, \blacklozenge \, 3) \, \bigstar \, 6 &= \left(5^2-3^2) \, \bigstar \, 6 \\
+(5 \, \blacklozenge \, 3) \, \bigstar \, 6 &= \left(5^2-3^2\right) \, \bigstar \, 6 \\
 &= 16 \, \bigstar \, 6 \\
 &= (16-6)^2 \\