AMC 10 · 2006 · #2
Grade 7 algebraPick an answer.
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.
Read the rule as difference of squares
The rule is just a difference of squares.
A sum times its matching difference always collapses to one square minus the other.
A sum multiplied by its matching difference always collapses to one square minus the other.
▸ Why?
A difference of two squares is exactly the two numbers added multiplied by the two numbers subtracted.
▸ Why?
Spreading each term of one bracket across the other makes the middle terms cancel and leaves only the squares.
Do the inner operation first
The inner operation gives -9.
Squaring the bigger second number makes the difference negative.
6.EE.A.1Identify SubproblemsFeed the result into the outer operation
Squaring erases its sign, so the outer step gives -72, choice (A).
Take away a much larger square from a small one and you land far below zero.
7.NS.A.1Eliminate PossibilitiesWhen 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