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

2006 AMC 12B Problem 2

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For real numbers

x

and

y,

define

x \spadesuit y = (x + y)(x - y).

What is

3 \spadesuit (4 \spadesuit 5)?

-72

-27

-24

24

72

Note that

x \spadesuit y = x^2 - y^2

Evaluate the inner

4 \spadesuit 5

first, then apply the operation again

Since

x \spadesuit y = x^2 - y^2,

the inner value is

4 \spadesuit 5 = 16 - 25 = -9.

Then

3 \spadesuit (-9) = 3^2 - (-9)^2

= 9 - 81 = -72.

Thus, the correct answer is A.