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Difference between revisions of "1989 AHSME Problems/Problem 18"

(Created page with "The set of all real numbers for which <cmath>x+\sqrt{x^2+1}-\frac{1}{x+\sqrt{x^2+1}}</cmath> is a rational number is the set of all (A) integers <math>x</math> (B) rational <m...")
 
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== Problem ==
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The set of all real numbers for which  
 
The set of all real numbers for which  
 
<cmath>x+\sqrt{x^2+1}-\frac{1}{x+\sqrt{x^2+1}}</cmath>
 
<cmath>x+\sqrt{x^2+1}-\frac{1}{x+\sqrt{x^2+1}}</cmath>
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(E) <math>x</math> for which <math>x+\sqrt{x^2+1}</math> is rational
 
(E) <math>x</math> for which <math>x+\sqrt{x^2+1}</math> is rational
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== Solution ==
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Rationalizing the denominator of <math>\frac{1}{x+\sqrt{x^2+1}}</math>, it simplifies: <math>\frac{1}{x+\sqrt{x^2+1}}</math> = <math>\frac{x-\sqrt{x^2+1}}{x^2-(x^2+1)}</math> = <math>\frac{x-\sqrt{x^2+1}}{-1}</math> = <math>-(x-\sqrt{x^2+1})</math>. Substituting this into the original equation, we get <math>x + \sqrt{x^2+1} - (-(x-\sqrt{x^2+1})) = x+\sqrt{x^2+1} + x - \sqrt{x^2+1} = 2x</math>. <math>2x</math> is only rational if <math>x</math> is rational <math>\mathrm{(B)}</math>
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== See also ==
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{{AHSME box|year=1989|num-b=17|num-a=19}} 
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[[Category: Intermediate  Algebra Problems]]
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{{MAA Notice}}

Latest revision as of 17:53, 30 September 2024

Problem

The set of all real numbers for which \[x+\sqrt{x^2+1}-\frac{1}{x+\sqrt{x^2+1}}\] is a rational number is the set of all

(A) integers $x$ (B) rational $x$ (C) real $x$

(D) $x$ for which $\sqrt{x^2+1}$ is rational

(E) $x$ for which $x+\sqrt{x^2+1}$ is rational

Solution

Rationalizing the denominator of $\frac{1}{x+\sqrt{x^2+1}}$, it simplifies: $\frac{1}{x+\sqrt{x^2+1}}$ = $\frac{x-\sqrt{x^2+1}}{x^2-(x^2+1)}$ = $\frac{x-\sqrt{x^2+1}}{-1}$ = $-(x-\sqrt{x^2+1})$. Substituting this into the original equation, we get $x + \sqrt{x^2+1} - (-(x-\sqrt{x^2+1})) = x+\sqrt{x^2+1} + x - \sqrt{x^2+1} = 2x$. $2x$ is only rational if $x$ is rational $\mathrm{(B)}$

See also

1989 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
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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