AMC 10 · 2005 · #9
Grade 8 algebraPick an answer.
First tidy the equation into standard form 4x²+(a+8)x+9=0. The key is Tool #5 (Look for a Pattern): a quadratic set equal to 0 has only one solution exactly when its left side is a perfect square, and the outer terms 4x² and 9 are already the squares of 2x and 3. Matching to the pattern (2x± 3)²=4x²± 12x+9 pins down the middle coefficient. Tool #4 (Introduce a Variable) then turns that match into the small equations a+8=± 12 and solves them for a; Tool #3 (Eliminate Possibilities) confirms the sum against the answer list.
Combine the two x-terms
Combining like terms puts the parameter into one coefficient.
Terms with the same variable part combine, so scattered x-terms collapse into one.
7.EE.A.1Introduce A VariableOne solution means perfect square
A single solution means the side is a perfect square.
A repeated root packs the quadratic into a single square, and 4x² and 9 announce which square to try.
Exactly one solution means the quadratic is a perfect square, and its outer terms announce which square to try.
▸ Why?
A quadratic vanishes only where one of its factors vanishes, so a single root means the two factors are the same.
▸ Why?
Two expressions equal for every input have equal coefficients term by term, which pins the middle term down.
Match the middle term
The ends are already squares, so only the middle needs matching, with two signs.
Two equal quadratics must match coefficient by coefficient, so the middle terms line up.
6.EE.A.3Introduce A VariableSolve for each value of a
Each sign gives one value, 4 and -20.
Undo the +8 once for each sign to peel off a.
8.EE.C.7Introduce A VariableAdd the two values
Adding them gives -16, choice (A).
The two values sit symmetrically around -8, so their sum lands at 2×(-8)=-16.
7.NS.A.1Eliminate PossibilitiesA quadratic has just one solution when its left side is a perfect square, so match it to (2x± 3)² and read off the middle term.
- Combine the two x-terms
- One solution means perfect square
- Match the middle term
- Solve for each value of a
- Add the two values