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2006 AMC 10B Problem 2

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2006 AMC 10B Problem 2AMC 10 Step by Step
Name:                             Date: October 28, 2026
Functions · difficulty 1 of 5 · about 1 min · position P1-10

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

  1. A)−72-72
  2. B)−27-27
  3. C)−24-24
  4. D)2424
  5. E)7272

Answer

A

Key insight

The spade rule is just the first number squared minus the second number squared. Work out the inside part first, then use its answer, sign included.

Solution

1. Make the rule simpler first. (x+y)(x−y)=x2−y2(x+y)(x-y) = x^2 - y^2, so x♠yx \spadesuit y means "square the first number, square the second, and subtract."
2. Parentheses go first, so start with the inside:
4♠5=42−52=16−25=−9.4 \spadesuit 5 = 4^2 - 5^2 = 16 - 25 = -9.
3. Now the problem is 3♠(−9)3 \spadesuit (-9). The first number is 33 and the second is −9-9:
32−(−9)2=9−81=−72.3^2 - (-9)^2 = 9 - 81 = -72.
Keep the order: it is the first number's square minus the second's, so the answer is negative.

The answer is (A) −72\boxed{\textbf{(A)}\ -72}.

Why this works

A made-up symbol is just a rule with two inputs. Rewriting it in a familiar form (x2−y2x^2 - y^2) makes it quicker to use. When one use of the rule sits inside another, work from the inside out, and carry the inside answer, sign and all, into the outer step.

Alternative approach

Use the rule exactly as written, with no rewriting. Inside: (4+5)(4−5)=9⋅(−1)=−9(4+5)(4-5) = 9 \cdot (-1) = -9. Outside: (3+(−9))(3−(−9))=(−6)(12)=−72(3 + (-9))(3 - (-9)) = (-6)(12) = -72.

The trap

Subtracting in the wrong order at the end, 81 - 9 = 72, instead of the first square minus the second, 9 - 81 = -72.

Concise solution

By the difference of squares, (x+y)(x−y)=x2−y2(x+y)(x-y) = x^2 - y^2, so x♠yx \spadesuit y is simply x2−y2x^2 - y^2.

Work from the inside out. First
4♠5=42−52=16−25=−9.4 \spadesuit 5 = 4^2 - 5^2 = 16 - 25 = -9.
Then
3♠(−9)=32−(−9)2=9−81=−72.3 \spadesuit (-9) = 3^2 - (-9)^2 = 9 - 81 = -72.

The answer is (A) −72\boxed{\textbf{(A)}\ -72}.

Review in the AoPS books

Techniques

Set up the equation/formula and compute; no special trick needed