AMC 10 · 2015 · #14

Grade 8 algebra
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
quadratic-equationsvieta-formulasoptimization convert-to-algebra ↑ Prerequisites: quadratic-equationsvieta-formulas
📏 Medium solution 💡 2 insights
Problem
Pick three different one-digit numbers a, b, c and plug them into the equation (x-a)(x-b)+(x-b)(x-c)=0. That equation has two roots. We want to choose a, b, c so that the two roots add up to the largest possible total, and report that total.

Pick an answer.

(A)
15
(B)
15.5
(C)
16
(D)
16.5
(E)
17

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

How to solve
Strategy Extreme Principle

The question asks for a maximum, so tool #14 (Extreme Principle) is the spine: once we have a clean formula for the sum of the roots, we push a, b, c to their largest allowed values. To get that clean formula we first lean on tool #4 (Introduce a Variable): instead of multiplying everything out, we keep a, b, c as letters, factor the shared (x-b), and read the roots straight off. The sum of the roots turns out to weight b more heavily than a and c, which tells the Extreme Principle exactly where to put the biggest number. Finally tool #3 (Eliminate Possibilities) matches the computed total to one of the five numeric choices.

1STEP 1

Factor out the shared piece

Both terms share (x-b), so factor it out; inside the bracket you are left with (x-a)+(x-c)=2x-a-c.

(x-b)[(x-a)+(x-c)]=(x-b)(2x-a-c)=0
2STEP 2

Read off the two roots

A product is zero only if a factor is zero: x-b=0 gives x=b, and 2x-a-c=0 gives x=(a+c)/2.

x=b or 2x-a-c=0 → x=(a+c)/2
3STEP 3

Write the sum of the roots

Adding the roots gives the total to maximize, b+(a+c)/2; here b counts fully while a and c each count only half.

sum=b+(a+c)/2
4STEP 4

Assign the biggest values and compute

Put the biggest in the full-weight slot b=9, then a=8, c=7, giving 9+(8+7)/2=16.5, choice (D).

b=9, a=8, c=7: 9+15/2=16.5 → (D)
Answer
16.5
Test a rival assignment to be sure b=9 really wins. If instead b=8 with a=9, c=7, the sum is 8+(9+7)/2=8+8=16, which is smaller than 16.5. Putting the 9 into a half-weight slot wastes part of it, confirming that the full-weight slot b should hold the largest value. So 16.5 is genuinely the maximum, and it sits right at the top of the answer choices, which is exactly where a maximum should land.
💡Key takeaway

Factor the shared piece to find the roots, then notice which variable counts the most in the sum and feed it your biggest number.

  • Factor out the shared piece
  • Read off the two roots
  • Write the sum of the roots
  • Assign the biggest values and compute

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