AMC 10 · 2006 · #2

Grade 7 algebra

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

function-evaluationdifference-of-squaresorder-of-operations identify-subproblems ↑ Prerequisites: function-evaluation

📏 Short solution 💡 1 insight

Problem

A made-up operation ♠ is defined for real numbers by x ♠ y = (x+y)(x-y). Using this rule, find the value of 3 ♠ (4 ♠ 5).

Pick an answer.

(A)

-72

(B)

-27

(C)

-24

(D)

(E)

AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The expression is one ♠ wrapped around another, so Tool #7 (Identify Subproblems) says: solve the inner operation 4 ♠ 5 first, get a single number, then feed that number into the outer 3 ♠ ( ). Spotting that (x+y)(x-y) is the difference-of-squares pattern x²-y² turns each ♠ into a quick 'square minus square'. Tool #3 (Eliminate Possibilities) guards the main trap: the operation is not associative, so grabbing 3 and 4 first, or swapping which number is x, lands on one of the wrong choices.

1STEP 1

Read the rule as difference of squares

Multiply (x+y)(x-y) out: the cross terms +xy and -xy cancel, so every ♠ is just x² - y² — first number squared minus second squared.

x ♠ y = (x+y)(x-y) = x² - y²

2STEP 2

Do the inner operation first

Inside the parentheses x=4 and y=5, so 4 ♠ 5 = 16 - 25 = -9 — the larger square is the one being taken away.

4 ♠ 5 = 4² - 5² = 16 - 25 = -9

3STEP 3

Feed the result into the outer operation

Now 3 ♠ (-9) with x=3, y=-9: 3² - (-9)² = 9 - 81 = -72, choice (A) — squaring -9 gives +81, not -81.

3 ♠ (-9) = 3² - (-9)² = 9 - 81 = -72 → (A)

Answer

-72

Each ♠ is 'a square minus a square', and in the final step we subtract 81 from only 9, so a large negative number is expected — that immediately rules out the positive choices (D) 24 and (E) 72. Squaring -9 correctly as +81 (not -81) is the key move; treating it as -81 would give 9-(-81)=90, which is not even a choice, confirming the sign was handled right. The magnitude 72 also matches 81-9, so -72 is consistent.

💡Key takeaway

When a made-up symbol is defined by a formula, do the operation inside the parentheses first, then plug that number into the outer one — and watch the signs when a square gets subtracted.

Read the rule as difference of squares
Do the inner operation first
Feed the result into the outer operation

Read the rule as difference of squares

Do the inner operation first

Feed the result into the outer operation

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