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. \(± \frac{1}{√2}\)
• C. ±√3
• D. \(±\frac{ 1 }{√ 3}\)

Answer 1

The Correct Option is A

Let the number be \( x \). Its reciprocal is \( \frac{1}{x} \). According to the problem, the sum of the number and its reciprocal is thrice the difference of the number and its reciprocal.

Mathematically, this can be expressed as: 

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

Expanding the equation:

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

Bringing all terms to one side gives:

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

Which simplifies to:

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

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

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

Simplify and factor the quadratic equation:

\( 2x^2 = 4 \)

\( x^2 = 2 \)

Taking the square root of both sides gives:

\( x = ±\sqrt{2} \)

Hence, the number is \( ±\sqrt{2} \).