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Algebra Difficulty 4.7 AIME Find the answer

2. Given a=12+3a=\frac{1}{2+\sqrt{3}}. Then the value of a2−2a+1\sqrt{a^{2}-2 a+1} is:

Pick one

Solution

2. A.

Notice that a=12+3=2−3<1a=\frac{1}{2+\sqrt{3}}=2-\sqrt{3}<1.
Then a2−2a+1=(a−1)2\sqrt{a^{2}-2 a+1}=\sqrt{(a-1)^{2}} =1−a=3−1=1-a=\sqrt{3}-1.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.