AMC 10 · 2005 · #10

Grade 8 algebra
Archetype Pair-Sum Invariant └ Sub-archetype Uniform Pair-Sum Recovery
quadratic-equationsperfect-squares pattern-recognitionconvert-to-algebra ↑ Prerequisites: quadratic-equationsperfect-squarespolynomial-factoring
📏 Medium solution 💡 2 insights
Problem
The equation 4x²+ax+8x+9=0 has a parameter a. For exactly two special values of a, this equation has only one solution for x. Find the sum of those two values of a.

Pick an answer.

(A)
-16
(B)
-8
(C)
0
(D)
8
(E)
20

AMC 10 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Look for a Pattern

First tidy the equation into standard form 4x²+(a+8)x+9=0. The key is Tool #5 (Look for a Pattern): a quadratic set equal to 0 has only one solution exactly when its left side is a perfect square, and the outer terms 4x² and 9 are already the squares of 2x and 3. Matching to the pattern (2x± 3)²=4x²± 12x+9 pins down the middle coefficient. Tool #4 (Introduce a Variable) then turns that match into the small equations a+8=± 12 and solves them for a; Tool #3 (Eliminate Possibilities) confirms the sum against the answer list.

1STEP 1

Combine the two x-terms

Both middle terms carry x, so they merge: ax+8x=(a+8)x, giving 4x²+(a+8)x+9=0. Only that middle coefficient depends on a.

4x²+ax+8x+9=4x²+(a+8)x+9=0
2STEP 2

One solution means perfect square

One solution means the left side is a perfect square; since 4x²=(2x)² and 9=3², it must be (2x±3)² — only the sign is unknown.

4x²+(a+8)x+9=(2x± 3)²
3STEP 3

Match the middle term

Expand: (2x+3)²=4x²+12x+9 and (2x-3)²=4x²-12x+9. The ends already agree, so the middle ones must: a+8=±12.

(2x± 3)²=4x²± 12x+9 → a+8=± 12
4STEP 4

Solve for each value of a

Subtract 8 from each side: a+8=12 gives a=4, and a+8=-12 gives a=-20.

a+8=12→ a=4; a+8=-12→ a=-20
5STEP 5

Add the two values

The ask is the sum, so add them: 4+(-20)=-16, which is choice (A).

4+(-20)=-16 → (A)
Answer
-16
The two values a=4 and a=-20 are the ± 12 shifts of -8, so their sum must be 2×(-8)=-16 regardless of the exact shift — a good sign the algebra is consistent. Checking a=4: the equation becomes 4x²+12x+9=(2x+3)²=0, whose only solution is x=-3/2. Checking a=-20: it becomes 4x²-12x+9=(2x-3)²=0, whose only solution is x=3/2. Both genuinely have one solution, and 4+(-20)=-16 matches choice (A).
💡Key takeaway

A quadratic has just one solution when its left side is a perfect square, so match it to (2x±3)² and read off the middle term.

  • Combine the two x-terms
  • One solution means perfect square
  • Match the middle term
  • Solve for each value of a
  • Add the two values

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