Competition · AMC preparation · step 4 of 4
AMC 10 · 2022A · #6
Grade 8 algebraPick an answer.
AMC 10 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
This is a multiple-choice problem where every answer choice is a specific function of a. Tool #3 (Eliminate Possibilities) is the AMC headline move — pick one concrete negative number, evaluate the original expression, then test each choice. Tool #9 (Easier Related Problem) is what we lean on to pick the test value: instead of carrying a symbolically, set a = -1 (smallest absolute value that stays negative). Tool #6 (Guess and Check) is the verification step. The hard symbolic property √((a-1)²)=|a-1| never needs to be stated abstractly — a single numerical evaluation makes it obvious. If two choices happened to tie at a=-1, we would simply repeat with a=-2 to break the tie.
Try a = -1
Pick the simplest negative test value: a = -1. It satisfies the condition and keeps the arithmetic tiny.
Replacing the unknown a with a single concrete negative number turns a symbolic problem into a counting check.
6.NS.C.5Solve An Easier Related ProblemSimplify the inner root
Inner root first: a-1 = -2, so (a-1)² = 4, and the principal square root is √(4) = 2.
√(x²) always gives back the positive size, never the negative — the square already erased the minus sign.
A square root of a square always gives back the size, never the negative.
▸ Why?
A root undoes a power, but only up to the sign the square erased.
▸ Why?
A number and its opposite have the same square, so the square cannot tell them apart.
Take the absolute value
Plug into the outer absolute value: -1-2-2 = -5, and taking the absolute value gives |-5| = 5.
Absolute value strips the sign — the distance from 0 on the number line.
6.NS.C.7Solve An Easier Related ProblemTest each answer choice
Evaluate each choice at a=-1, keeping those equal to 5: (A) 3-2(-1)=5, (B) 2, (C) 1, (D) 0, (E) 3. Only (A) matches.
Substitute the same number into every choice; the right one returns the same value as the original.
6.EE.A.2Eliminate PossibilitiesRecheck with a = -2
Confirm at a=-2: √((-3)²)=3, then |-2-2-3| = |-7| = 7, and (A) 3-2(-2)=7 matches again; no other choice does.
One match could be coincidence; two matches in a row across different inputs confirm the algebraic identity.
6.EE.A.4Guess And CheckThis AMC 10 problem only needs Grade 8 square-root reasoning you already know — plug in a=-1, follow the order of operations, and test each choice; the one that lands on the same number is (A) 3-2a.
- Try a = -1
- Simplify the inner root
- Take the absolute value
- Test each answer choice
- Recheck with a = -2
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