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2022 AMC 10A Problem 6

Problem 6 of 25EasierAlgebra

Which expression is equal to ∣a−2−(a−1)2∣\left|a-2-\sqrt{(a-1)^2}\right| for a<0?a < 0?

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Solution

By the definition of square root, we get that (a−1)2=∣a−1∣.\sqrt{(a - 1)^2} = |a - 1|. Since a<0,a < 0, we get that a−1<0,a - 1 < 0, which means that ∣a−1∣=1−a.|a - 1| = 1 - a. The whole expression therefore simplifies to ∣a−2−(1−a)∣=∣2a−3∣. |a - 2 - (1 - a)| = |2a - 3|. Since a<0,a < 0, we know that 2a−3<0.2a - 3 < 0. This means that ∣2a−3∣=3−2a.|2a - 3| = 3 - 2a. Thus, A is the correct answer.

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Concepts: absolute value · radical

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.