AMC 10 · 2003 · #5
Grade 8 algebraPick an answer.
AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The target (d-1)(e-1) expands into de-(d+e)+1, so the only two facts it needs are the sum d+e and the product de of the roots — that is tool #16 (Change Focus): stop trying to find each root and focus instead on their sum and product. Tool #4 (Introduce a Variable) supplies those two facts cheaply: a quadratic 2x²+3x-5 with roots d,e must equal 2(x-d)(x-e), and expanding that and matching coefficients reads the sum and product straight off b and c. Tool #6 (Guess and Check) gives a fast sanity route: testing x=1 shows 2+3-5=0, so 1 is actually one of the roots, which forces a factor of (1-1)=0 into the product — a quick confirmation of the answer.
Expand the target product
Expanding gives (d-1)(e-1)=de-d-e+1=de-(d+e)+1, so only the roots' product and sum matter.
Expanding first shows the answer only needs the roots' sum and product, so there is no reason to hunt down each root.
6.EE.A.3Change Focus Count The ComplementRead sum and product off the coefficients
Matching 2(x-d)(x-e)=2x²-2(d+e)x+2de against 2x²+3x-5 gives d+e=-3/2 and de=-5/2.
Writing the quadratic as its factored form makes the sum and product of the roots appear directly as the middle and constant coefficients.
Writing the quadratic in factored form makes the sum and product of its roots appear as its coefficients.
▸ 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 factored form is honest.
Substitute and simplify
Putting de=-5/2 and d+e=-3/2 into de-(d+e)+1 gives -5/2+3/2+1, so -1+1=0 — choice (B).
Once the sum and product are known, the answer is just a short piece of signed-fraction arithmetic.
7.NS.A.1Introduce A VariableTo evaluate an expression in both roots of a quadratic, expand it into the roots' sum and product, which you can read straight off the coefficients — you rarely need to find the roots themselves.
- Expand the target product
- Read sum and product off the coefficients
- Substitute and simplify