AMC 10 · 2013 · #19

Grade 8 algebra

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

sequences-arithmeticquadratic-equationscompleting-the-square convert-to-algebracompleting-the-square ↑ Prerequisites: quadratic-equationssequences-arithmetic

📏 Long solution 💡 3 insights

Problem

Three real numbers written in the order c, b, a form an arithmetic sequence, and they satisfy a ≥ b ≥ c ≥ 0. The quadratic ax² + bx + c has exactly one root. Find that root.

Pick an answer.

(A)

$-7-4\sqrt{3}$

(B)

$-2-\sqrt{3}$

(C)

-1

(D)

$-2+\sqrt{3}$

(E)

$-7+4\sqrt{3}$

AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Convert to Algebra

Both facts are verbal: 'arithmetic sequence' and 'exactly one root'. Turn each into an equation, then combine them. Scaling by a shrinks three unknowns to two ratios, so the system collapses to one quadratic in a single variable. Finally the ordering a ≥ b ≥ c picks which of the two solutions is legal.

1STEP 1

Turn the sequence into an equation

Equal gaps mean b - c = a - b, so 2b = a + c: the middle term is the average of its neighbors.

b - c = a - b → 2b = a + c

2STEP 2

Write what 'one root' means

One root forces the perfect square a(x - r)², and matching coefficients gives b² = 4ac with r = -b/(2a).

a(x-r)² = ax² - 2ar x + ar² → b² = 4ac, r = -b/2a

3STEP 3

Scale so a = 1

Dividing by a keeps the root, so with p = b/a and q = c/a the conditions read 2p = 1 + q and p² = 4q, and the root is -p/2.

p = b/a, q = c/a: 2p = 1 + q, p² = 4q, r = -p/2

4STEP 4

Combine into one equation

Substitute q = 2p - 1 into p² = 4q to erase q, which leaves p² - 8p + 4 = 0.

q = 2p - 1 → p² = 8p - 4 → p² - 8p + 4 = 0

5STEP 5

Solve by completing the square

Adding 16 to both sides makes (p - 4)² = 12, so p - 4 = ±2√(3) and p = 4 ± 2√(3).

(p-4)² = 12 → p = 4 ± 2√(3)

6STEP 6

Pick the valid ratio

Since a ≥ b forces p ≤ 1, the option near 7.46 is out and p = 4 - 2√(3) ≈ 0.54 survives, giving q ≈ 0.07.

p ≤ 1 → p = 4 - 2√(3) ≈ 0.54, q = 7 - 4√(3) ≈ 0.07

7STEP 7

Read off the root

Halving p and flipping the sign gives the single root r = -p/2 = -2 + √(3), which is answer (D).

r = -p/2 = -(4 - 2√(3))/2 = -2 + √(3)

Answer

-2+√(3)

Take a = 1, so p = 4 - 2√(3) gives b = 4 - 2√(3) ≈ 0.54 and c = q = 7 - 4√(3) ≈ 0.07. Then 1 ≥ 0.54 ≥ 0.07 ≥ 0 holds, and c, b, a = 0.07, 0.54, 1 are evenly spaced (gap ≈ 0.46). The discriminant b² - 4ac = (4 - 2√(3))² - 4(7 - 4√(3)) = (28 - 16√(3)) - (28 - 16√(3)) = 0, confirming exactly one root at -b/(2a) = -2 + √(3) ≈ -0.27.

💡Key takeaway

Turn each word-fact into an equation, combine them into one quadratic, then let the size rule a ≥ b ≥ c choose the answer that fits.

Turn the sequence into an equation
Write what 'one root' means
Scale so a = 1
Combine into one equation
Solve by completing the square
Pick the valid ratio
Read off the root