AMC 10 · 2015 · #12
Grade 8 algebraPick an answer.
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.
Factor out the shared piece
A shared factor makes it factor at once.
A common factor sitting in every term can always be pulled out front, which exposes the structure hiding underneath.
6.EE.A.3Introduce A VariableRead off the two roots
One root is a digit; the other is an average.
Zero can only come out of a product if one factor is already zero, so each factor hands you one root.
Zero can only come out of a product if one factor is already zero, so each factor hands you one root.
▸ Why?
Two nonzero numbers can never multiply to zero, so no root can hide outside the factors.
▸ Why?
Pulling the shared piece out front is what turns the sum into a product in the first place.
Write the sum of the roots
So one slot carries twice the weight.
When one variable carries more weight in the sum, spending your biggest number there buys the most.
6.EE.A.2Extreme PrincipleAssign the biggest values and compute
Assigning the biggest digits gives 16.5, choice (D).
Largest number in the heaviest slot, next-largest in the rest — that is how an extreme value is built.
6.EE.A.2Eliminate PossibilitiesFactor 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