AMC 10 · 2015 · #23

Grade 8 arithmetic

Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration

vieta-formulasquadratic-equationspolynomial-roots simons-favorite-factoring-trick ↑ Prerequisites: quadratic-equationsvieta-formulas

📏 Medium solution 💡 3 insights

Problem

The quadratic f(x)=x²-ax+2a has two zeroes that are both integers. Add up every value of a that makes this happen.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

Tool #4 (Introduce a Variable) names the two integer roots p and q; matching f(x)=(x-p)(x-q) to x²-ax+2a turns the word 'zeroes are integers' into two clean equations p+q=a and pq=2a. Tool #15 (Organize Information in More Ways) rearranges the leftover relation pq=2(p+q) into the factored form (p-2)(q-2)=4, which is far easier to search than a scattered equation. Tool #2 (Make a Systematic List) then sweeps every integer factor pair of 4 — including the negatives, the easy ones to forget — so no value of a is missed.

1STEP 1

Name the integer roots

Naming the roots p,q and matching coefficients to x²-ax+2a gives p+q=a and pq=2a.

f(x)=(x-p)(x-q)=x²-(p+q)x+pq → p+q=a, pq=2a

2STEP 2

Eliminate a

Substituting a=p+q into pq=2a eliminates a, leaving one relation between the roots: pq=2(p+q).

pq=2a=2(p+q)

3STEP 3

Factor into a product equal to 4

Adding 4 to both sides makes the left side factor neatly: (p-2)(q-2)=4.

pq-2p-2q=0 → pq-2p-2q+4=4 → (p-2)(q-2)=4

4STEP 4

List the factor pairs of 4

Listing all integer factor pairs of 4 — including the easy-to-miss negatives — recovers every possible value of a.

(1,4) :a=9; (2,2) :a=8; (-1,-4) :a=-1; (-2,-2) :a=0

5STEP 5

Add the distinct values of a

Adding the four distinct values of a gives 9+8+(-1)+0=16, choice (C).

9+8+(-1)+0=16 → (C)

Answer

Spot-check one value: for a=-1, f(x)=x²+x-2=(x+2)(x-1), roots -2 and 1 — both integers, good. For a=0, f(x)=x² has the integer double root 0. The four values 9,8,-1,0 are exactly the four factor pairs of 4, so the list is complete. Their sum 16 is choice (C); the popular wrong answer 17 (D) is precisely what you get by dropping the two negative factor pairs and keeping only 9+8.

💡Key takeaway

Name the roots, turn the conditions into (p-2)(q-2)=4, then list every factor pair of 4 — negatives included — so no a slips away.

Name the integer roots
Eliminate a
Factor into a product equal to 4
List the factor pairs of 4
Add the distinct values of a

Name the integer roots

Eliminate a

Factor into a product equal to 4

List the factor pairs of 4

Add the distinct values of a

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