AMC 10 · 2015 · #18
Grade 8 algebranumber-theoryPick an answer.
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.
Name the integer roots
Two roots give the sum and the product.
A monic quadratic is fully described by its two roots, so the coefficients are just their sum and product.
A monic quadratic is fully described by its two roots, so its coefficients are just their sum and product.
▸ Why?
The coefficients record exactly the sum and the product of the roots, and nothing else.
▸ Why?
A quadratic vanishes only where one of its linear factors vanishes, so the roots really do rebuild it.
Eliminate a
Eliminating the coefficient leaves one equation.
Two equations sharing one unknown collapse to one equation once you swap that unknown out.
8.EE.C.8Introduce A VariableFactor into a product equal to 4
Adding a constant makes it a plain product.
Padding the expression by a constant rewrites a messy equation as 'two factors multiply to a fixed number.'
6.EE.A.3Organize Information In More WaysList the factor pairs of 4
There are only four factor pairs.
A product of 4 comes from a short, checkable list of integer factor pairs — and the negative pairs count too.
7.NS.A.2Make A Systematic ListAdd the distinct values of a
Adding the values gives 16, choice (C).
Summing a finite list of integers is exact once you are sure the list is complete.
7.NS.A.1Make A Systematic ListName the roots, turn the conditions into (p-2)(q-2)=4, then list every factor pair of 4 — negatives included — so no value of 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