If y=√x^2-2x+1 +√x^2+2x+1, then y is:

1957 AMC 12 Problem 14

Attempt Problem 14 of the 1957 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1957 AMC 12 solutions, or check the answer key.

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14.

y=\sqrt{x^2-2x+1}

{}+\sqrt{x^2+2x+1},

then

y

is:

2x

2(x+1)

0

|x-1|+|x+1|

none of these

Factor each radicand as a perfect square

For real

u,

\sqrt{u^2}=|u|

The two radicands are

(x-1)^2

and

(x+1)^2.

Hence

\begin{aligned} y&=\sqrt{(x-1)^2}\\ &\quad+\sqrt{(x+1)^2}\\ &=|x-1|+|x+1|. \end{aligned}

Thus, the correct answer is D.