The set of all real numbers x for which x+√x^2+1-1x+√x^2+1 is a rational number is the set of all

(D) x for which √x^2+1 is rational

(E) x for which x+√x^2+1 is rational

1989 AMC 12 Problem 18

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18.

The set of all real numbers

x

for which

x+\sqrt{x^2+1}-\frac1{x+\sqrt{x^2+1}}

is a rational number is the set of all

integers

x

rational

x

real

x

x

for which

\sqrt{x^2+1}

is rational

x

for which

x+\sqrt{x^2+1}

is rational

Rationalize the reciprocal term

The conjugate

\sqrt{x^2+1}-x

is the reciprocal of the denominator

Rationalizing gives

\frac1{x+\sqrt{x^2+1}}=\sqrt{x^2+1}-x.

The entire expression is therefore

2x.

It is rational exactly when

x

is rational.

Thus the correct answer is B.