AMC 10 · 2022 · #4

Grade 7 algebra

Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In

polynomial-factoringpolynomial-rootsvieta-formulasfactorssystematic-enumeration systematic-enumerationpattern-recognitionidentify-subproblems ↑ Prerequisites: polynomial-factoringfactors

📏 Medium solution 💡 2 insights

Problem

A quadratic carries one constant coefficient. Find how many integer values of that constant give the quadratic two distinct integer roots.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

How to solve

Strategy Make a Systematic List

The roots have to multiply to 36, so we are really being asked to list every integer factor pair of 36 in which the two factors are different. Tool #2 (Systematic List) is built for 'find every pair' with no duplicates. Tool #7 (Subproblems) splits the work into 'list positive factor pairs', 'list negative factor pairs', and 'count distinct k'. Tool #9 (Easier Problem) is the bridge: instead of solving for k first, solve the simpler factor-pair question and read k off at the end. We avoid the quadratic formula — factor-pair counting is the natural elementary path.

1STEP 1

Write the root relations

The roots' product is fixed.

x² + kx + 36 = (x-r)(x-s) → rs = 36, r + s = -k

2STEP 2

List the positive pairs

List the positive factor pairs.

positive pairs with r ＜ s: (1, 36), (2, 18), (3, 12), (4, 9)

3STEP 3

List the negative pairs

The sign-flipped pairs count too.

negative pairs: (-1, -36), (-2, -18), (-3, -12), (-4, -9)

4STEP 4

Kill the mixed-sign pairs

Mixed signs make the product negative.

(+) × (-) = (-) ≠ 36

5STEP 5

Collect the constants

Each pair gives one constant.

k ∈ {-37, -20, -15, -13, 37, 20, 15, 13}

6STEP 6

Count them

There are 8 of them.

|{-37, -20, -15, -13, 13, 15, 20, 37}| = 8 → (B)

Answer

Spot-verify the largest k: if k = -37 the polynomial is x² - 37x + 36 = (x - 1)(x - 36) with roots 1 and 36 ✓. Spot-verify a positive k: if k = 13 then x² + 13x + 36 = (x + 4)(x + 9) with roots -4 and -9 ✓. The structural reason the count is 8 and not 9: the factor pair (6, 6) — which would give k = -12 — is forbidden because the roots must be distinct. The structural reason it is not 14 or 16: pairs like (1, 36) and (36, 1) are the same pair (and give the same k), so unordered pair counting is correct.

💡Key takeaway

This AMC 12 problem only needs Grade 7 expanding (x-r)(x-s) you already know — the constant 36 has to come from r · s, so list the four positive factor pairs of 36 with r ≠ s, double them for negatives, and read off 8 values of k.