1957 AHSME Problems/Problem 22

Problem

If $\sqrt{x - 1} - \sqrt{x + 1} + 1 = 0$, then $4x$ equals:

$\textbf{(A)}\ 5 \qquad \textbf{(B)}\ 4\sqrt{-1}\qquad \textbf{(C)}\ 0\qquad \textbf{(D)}\ 1\frac{1}{4}\qquad \textbf{(E)}\ \text{no real value}$

Solution

By repeatedly rearranging the equation and squaring both sides, we can solve for $x$: \begin{align*} \sqrt{x-1}-\sqrt{x+1}+1 &= 0 \\ \sqrt{x-1}+1 &= \sqrt{x+1} \\ x-1+2\sqrt{x-1}+1 &= x+1 \\ 2\sqrt{x-1} &= 1 \\ \sqrt{x-1} &= \frac{1}{2} \\ x-1 &= \frac{1}{4} \\ x &= \frac{5}{4} \end{align*} After checking for extraneous solutions, we see that $x=\tfrac{5}{4}$ does indeed solve the equation. Thus, $4x=5$, and so our answer is $\boxed{\textbf{(A) }5}$.

See Also

1957 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 21
Followed by
Problem 23
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