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2006 AMC 12B Problem 2

Problem 2 of 25EasierAlgebra

For real numbers xx and y,y, define x♠y=(x+y)(x−y).x \spadesuit y = (x + y)(x - y). What is 3♠(4♠5)?3 \spadesuit (4 \spadesuit 5)?

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Solution

Since x♠y=x2−y2,x \spadesuit y = x^2 - y^2, the inner value is 4♠5=16−25=−9.4 \spadesuit 5 = 16 - 25 = -9. Then 3♠(−9)=32−(−9)23 \spadesuit (-9) = 3^2 - (-9)^2 =9−81=−72.= 9 - 81 = -72. Thus, the correct answer is A.
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