AMC 10 · 2014 · #20
Grade 8 arithmeticPick an answer.
AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The expression only uses x⁴ and x², so letting y=x² (Tool #4) turns a scary quartic into an ordinary quadratic y²-51y+50 that factors easily. Tool #7 (Identify Subproblems) then splits the job: first factor, then decide when the product is negative, then translate that back into a range for x. Tool #2 (Make a Systematic List) finishes by counting the integers in that range, remembering each positive value has a matching negative one.
Substitute to get a quadratic
Every power of x is even, so let y=x². Then x⁴ becomes y², and the quartic turns into the quadratic y²-51y+50.
Naming the repeated piece x² as one variable hides the scary exponent and leaves an ordinary quadratic.
6.EE.B.6Introduce A VariableFactor and undo the substitution
Two numbers multiply to 50 and add to 51: 1 and 50. So y²-51y+50=(y-1)(y-50), and undoing y=x² gives (x²-1)(x²-50).
Factoring rewrites the sum-shaped expression as a product, and a product's sign is easy to control.
Factoring rewrites the sum as a product, and a product's sign is easy to control.
▸ Why?
Opening the product sends each piece against each piece, so the factoring can be checked directly.
▸ Why?
A product changes sign only where one of its factors does, so the cases are read off the factors.
Decide when the product is negative
A product is negative only with opposite signs, and x²-1 is always the larger, so x²-1 must be positive while x²-50 is negative.
For a product to dip below zero the factors must fight — one pushing positive, one pushing negative.
7.NS.A.2Identify SubproblemsTurn the two conditions into a range
So 1 ＜ x² ＜ 50. Since 7²=49 is under 50 but 8²=64 is over, the size of x runs 2 through 7.
The perfect squares strictly between 1 and 50 are 4,9,16,25,36,49, which pin down exactly which whole-number sizes of x work.
8.EE.A.2Identify SubproblemsCount both signs
Each size gives a positive and a negative twin, so 2,3,4,5,6,7 plus their negatives make 6+6=12 integers — choice (C).
Because only even powers appear, every positive solution is mirrored by its negative twin, so count one side and double it.
6.EE.B.5Make A Systematic ListRename x² as one letter, factor into two pieces, and the answer is just the integers whose square lands strictly between 1 and 50 — counted for both signs.
- Substitute to get a quadratic
- Factor and undo the substitution
- Decide when the product is negative
- Turn the two conditions into a range
- Count both signs