1957 AHSME Problems/Problem 14

If $y = \sqrt{x^2 - 2x + 1} + \sqrt{x^2 + 2x + 1}$, then $y$ is:

$\textbf{(A)}\ 2x\qquad \textbf{(B)}\ 2(x+1)\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ |x-1|+|x+1|\qquad\textbf{(E)}\ \text{none of these}$

Solution

In order to solve the problem, we will use two properties, namely, that $(a+b)^2=a^2+2ab+b^2$ and $(a-b)^2=a^2-2ab+b^2$.

We can use this to simplify the equation, as $y = \sqrt{x^2 - 2x + 1} + \sqrt{x^2 + 2x + 1}$ turns into $y = x + 1 + x - 1$. However, the square root function only allows for nonnegative inputs and only generates nonnegative outputs, so $y$ is $\boxed{\textbf{(D) }|x+1|+|x-1|}$.

See Also

1957 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 13
Followed by
Problem 15
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