AMC 10 · 2002 · #6
Grade 8 algebraPick an answer.
The whole sentence 'the roots are a and b' is a word fact, so Tool #13 (Convert to Algebra) turns it into symbols: rebuild the quadratic in factored form and match it term-by-term to x²+ax+b, which produces two equations linking a and b. Tool #3 (Eliminate Possibilities) then uses the 'nonzero' rule to throw away a stray branch when one equation could split two ways. Tool #6 (Guess and Check) confirms the winning pair by plugging it back and reading off the roots.
Rebuild the quadratic from its roots
Rebuilding from the roots gives a second form: x² - (a+b)x + ab.
If you know where a parabola hits zero, you can rebuild its equation by multiplying the two 'x-root' pieces.
6.EE.A.3Convert To AlgebraMatch the like parts
Matching term by term gives the pair a = -(a+b) and b = ab.
If two expressions are equal for every x, the number in front of each power has to match.
Two expressions equal for every x must carry the same number in front of each power.
▸ Why?
Polynomials that agree everywhere agree coefficient by coefficient, so one identity becomes a list of plain equations.
▸ Why?
Rebuilding the quadratic from its roots is what puts the roots into the coefficients in the first place.
Apply the nonzero rule to the constant equation
Factoring b(a-1)=0 and using the nonzero promise forces a = 1.
A product is zero only when one factor is zero, and the 'nonzero' promise tells you which factor it must be.
8.EE.C.7Eliminate PossibilitiesSolve for b and read off the pair
Substituting back gives b = -2, so the pair is (1,-2), choice (C).
With one unknown pinned down, the leftover equation hands you the other in a single step.
8.EE.C.7Convert To AlgebraIf a quadratic's own roots are its coefficients, rebuild it as (x-a)(x-b), match the pieces to x²+ax+b, and let the 'nonzero' rule pick the real answer.
- Rebuild the quadratic from its roots
- Match the like parts
- Apply the nonzero rule to the constant equation
- Solve for b and read off the pair