2022 AMC 10A Problem 6

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