Problem: Which expression is equal to
∣a−2−(a−1)2∣\left|a-2-\sqrt{(a-1)^{2}}\right| ∣∣∣∣​a−2−(a−1)2​∣∣∣∣​
for a<0a<0a<0?
Answer Choices:
A. 3−2a3-2 a3−2a B. 1−a1-a1−a C. 111 D. a+1a+1a+1 E. 333
Solution:
Because aaa is negative, a−1<0a-1<0a−1<0, so (a−1)2=∣a−1∣=1−a\sqrt{(a-1)^{2}}=|a-1|=1-a(a−1)2​=∣a−1∣=1−a. Therefore distributing gives a−2−(1−a)=2a−3a-2-(1-a)=2 a-3a−2−(1−a)=2a−3. Because aaa is negative, 2a−3<02 a-3<02a−3<0, so ∣2a−3∣=(A)3−2a|2 a-3|=(\text{A})\boxed{3-2a}∣2a−3∣=(A)3−2a​.
The problems on this page are the property of the MAA's American Mathematics Competitions