Difference between revisions of "1991 AHSME Problems/Problem 27"
(Created page with "If <math>x+\sqrt{x^2-1}+\frac{1}{x-\sqrt{x^2-1}}=20</math> then <math>x^2+\sqrt{x^4-1}+\frac{1}{x^2+\sqrt{x^4-1}}=</math> (A) <math>5.05</math> (B) <math>20</math> (C) <math>51....") |
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− | + | == Problem == | |
− | + | If <cmath>x+\sqrt{x^2-1}+\frac{1}{x-\sqrt{x^2-1}}=20,</cmath> then <cmath>x^2+\sqrt{x^4-1}+\frac{1}{x^2+\sqrt{x^4-1}}=</cmath> | |
+ | |||
+ | <math>\textbf{(A) } 5.05 \qquad | ||
+ | \textbf{(B) } 20 \qquad | ||
+ | \textbf{(C) } 51.005 \qquad | ||
+ | \textbf{(D) } 61.25 \qquad | ||
+ | \textbf{(E) } 400</math> | ||
+ | |||
+ | == Solution == | ||
+ | We rationalize the denominator in the given equation, then solve for <math>x:</math> | ||
+ | <cmath>\begin{align*} | ||
+ | x+\sqrt{x^2-1}+\frac{x+\sqrt{x^2-1}}{\left(x-\sqrt{x^2-1}\right)\left(x+\sqrt{x^2-1}\right)} &= 20 \\ | ||
+ | x+\sqrt{x^2-1}+x+\sqrt{x^2-1} &= 20 \\ | ||
+ | x+\sqrt{x^2-1} &= 10 \\ | ||
+ | \sqrt{x^2-1} &= 10-x \\ | ||
+ | x^2-1 &= 100-20x+x^2 \\ | ||
+ | 20x &= 101 \\ | ||
+ | x &= 5.05. | ||
+ | \end{align*}</cmath> | ||
+ | We rationalize the denominator in the requested expression, then simplify the result: | ||
+ | <cmath>\begin{align*} | ||
+ | x^2+\sqrt{x^4-1}+\frac{1}{x^2+\sqrt{x^4-1}} &= x^2+\sqrt{x^4-1}+\frac{x^2-\sqrt{x^4-1}}{\left(x^2+\sqrt{x^4-1}\right)\left(x^2-\sqrt{x^4-1}\right)} \\ | ||
+ | &= x^2+\sqrt{x^4-1}+x^2-\sqrt{x^4-1} \\ | ||
+ | &= 2x^2 \\ | ||
+ | &= \boxed{\textbf{(C) } 51.005}. | ||
+ | \end{align*}</cmath> | ||
+ | |||
+ | ~Hapaxoromenon (Solution) | ||
+ | |||
+ | ~MRENTHUSIASM (Reformatting) | ||
+ | |||
+ | == See also == | ||
+ | {{AHSME box|year=1991|num-b=26|num-a=28}} | ||
+ | |||
+ | [[Category: Intermediate Algebra Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 05:26, 6 September 2021
Problem
If then
Solution
We rationalize the denominator in the given equation, then solve for
We rationalize the denominator in the requested expression, then simplify the result:
~Hapaxoromenon (Solution)
~MRENTHUSIASM (Reformatting)
See also
1991 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 26 |
Followed by Problem 28 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
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