# 2022 AMC 10A Problems/Problem 6

## Problem

Which expression is equal to $$\left|a-2-\sqrt{(a-1)^2}\right|$$ for $a<0?$ $\textbf{(A) } 3-2a \qquad \textbf{(B) } 1-a \qquad \textbf{(C) } 1 \qquad \textbf{(D) } a+1 \qquad \textbf{(E) } 3$

## Solution 1

We have \begin{align*} \left|a-2-\sqrt{(a-1)^2}\right| &= \left|a-2-|a-1|\right| \\ &=\left|a-2-(1-a)\right| \\ &=\left|2a-3\right| \\ &=\boxed{\textbf{(A) } 3-2a}. \end{align*} ~MRENTHUSIASM

## Solution 2

Assume that $a=-1.$ Then, the given expression simplifies to $5$: \begin{align*} \left|a-2-\sqrt{(a-1)^2}\right| &= \left|-1-2-\sqrt{(-1-1)^2}\right| \\ &= \left|-1-2-\sqrt{4}\right| \\ &= \left|-1-2-2\right| \\ &= 5. \end{align*} Then, we test each of the answer choices to see which one is equal to $5$: $\textbf{(A) } 3-2a = 3-2\cdot(-1) = 3+2 = 5.$ $\textbf{(B) } 1-a = 1-(-1) = 2 \neq 5.$ $\textbf{(C) } 1 \neq 5.$ $\textbf{(D) } a+1 = -1+1 = 0 \neq 5.$ $\textbf{(E) } 3 \neq 5.$

The only answer choice equal to $5$ for $a=-1$ is $\boxed{\textbf{(A) } 3-2a}.$

-MathWizard09

## Video Solution 1 (Quick and Easy)

~Education, the Study of Everything

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 