1957 AHSME Problems/Problem 22
Problem
If 
, then 
equals:
Solution
By repeatedly rearranging the equation and squaring both sides, we can solve for 
:
\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 
does indeed solve the equation. Thus, 
, and so our answer is 
.
See Also
| 1957 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 21 |
Followed by Problem 23 | |
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