Question

The sum of a number and its reciprocal is thrice the difference of the number and its reciprocal. The number is:

• A. ±√2
• B. \(± \frac{1}{\sqrt2}\)
• C. \(± \frac{1}{3}\)
• D. \(± \sqrt3\)

Answer 1

The Correct Option is A

Let the number be \( x \). According to the problem, the sum of the number and its reciprocal is thrice the difference of the number and its reciprocal. The equation can be set as: 

\( x + \frac{1}{x} = 3(x - \frac{1}{x}) \)

Expanding the right-hand side, we get:

\( x + \frac{1}{x} = 3x - \frac{3}{x} \)

Rearranging the terms, we have:

\( x + \frac{1}{x} - 3x + \frac{3}{x} = 0 \)

Combine like terms:

\( -2x + \frac{4}{x} = 0 \)

Multiply through by \( x \) to eliminate the fraction:

\( -2x^2 + 4 = 0 \)

Rearranging gives us:

\( 2x^2 = 4 \)

Divide by 2:

\( x^2 = 2 \)

Taking the square root of both sides, we find:

\( x = \pm \sqrt{2} \)

Thus, the number is \( \pm \sqrt{2} \).