AMC 10 · 2022 · #6

Grade 8 algebra
absolute-valuesign-analysis caseworkeasier-related-problem ↑ Prerequisites: absolute-value
📏 Medium solution 💡 2 insights
Problem
Simplify the expression |a-2-√((a-1)²)| assuming a is a negative number. The answer must be one of five algebraic expressions in a.

Pick an answer.

(A)
3-2a
(B)
1-a
(C)
1
(D)
a+1
(E)
3

AMC 10 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Eliminate Possibilities

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.

1STEP 1

Pick the simplest negative test value: a = -1. It satisfies the condition and keeps the arithmetic tiny.

a = -1
2STEP 2

Inner root first: a-1 = -2, so (a-1)² = 4, and the principal square root is √(4) = 2.

√((a-1)²) = √((-2)²) = √(4) = 2
3STEP 3

Plug into the outer absolute value: -1-2-2 = -5, and taking the absolute value gives |-5| = 5.

|a - 2 - 2| = |-5| = 5
4STEP 4

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.

(A) 3-2a = 5 ✓ (B) 2 (C) 1 (D) 0 (E) 3
5STEP 5

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.

a=-2: |-2-2-3| = 7, 3-2(-2)=7 ✓
Answer
3-2a
Two independent test values a=-1 and a=-2 both produced exactly the value of 3-2a. Since the four other choices each disagreed at a=-1, only (A) can be the correct identity. As a sanity check on the structure: when a is negative, a-2-√((a-1)²) = a-2-(1-a) = 2a-3, which is negative, so taking its absolute value flips the sign to 3-2a — exactly choice (A).
💡Key takeaway

This 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.