AMC 10 · 2002 · #13
Grade 9 algebraPick an answer.
The property "differs from its reciprocal by 1" is a sentence about a single number, so tool #4 (Introduce a Variable) lets one letter x stand for either of a and b, and tool #13 (Convert to Algebra) turns the property into an equation. The one thing that must not be rushed is the word "differs": it hides an absolute value, so the honest translation is two equations, not one. Solving only x - 1/x = 1 would produce a single positive number and leave the problem with no second number at all. Tool #3 (Eliminate Possibilities) then does the filtering the problem asks for — four roots come out of the algebra and the word "positive" keeps exactly two. Tool #15 (Organize Information in More Ways) supplies the shorter second route used in the review.
Write the property as one condition
A gap of 1 means a distance, so both orders count: x - 1/x = 1 or 1/x - x = 1.
"Differ by 1" tells you how far apart, not which one is on top, so there are two directions to cover.
9.A-CED.A.1Introduce A VariableClear the reciprocal
Multiplying each through by x clears the reciprocal, giving x²-x-1 = 0 and x²+x-1 = 0.
Multiplying by x trades an awkward reciprocal for a plain quadratic.
9.A-SSE.A.2Convert To AlgebraKeep only the positive roots
Discarding the negative roots leaves the two positives (1+√5)/2 and (-1+√5)/2.
The algebra hands back four roots; the single word "positive" in the problem keeps two of them.
9.A-REI.B.4Eliminate PossibilitiesAdd the two numbers
Adding cancels the constants: 2√5/2 = √5, choice (C).
The two roots sit symmetrically on either side of √(5)/2, so their sum is just twice that midpoint.
The two surviving roots sit the same distance either side of one midpoint, so their sum is twice that midpoint.
▸ Why?
A quadratic's coefficients already carry the sum of its roots, so the total can be read off without finding either root.
▸ Why?
Pairing two values that lie symmetrically about a centre always gives twice that centre, whatever the spread.
"Differs by" hides an absolute value, so cover both directions — then let the word "positive" throw away the roots the problem never wanted.
- Write the property as one condition
- Clear the reciprocal
- Keep only the positive roots
- Add the two numbers