Question

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

• A. \(\pm\sqrt{2}\)
• B. \(\dfrac{1}{\pm\sqrt{2}}\)
• C. \(\pm\sqrt{3}\)
• D. \(\dfrac{1}{\pm\sqrt{3}}\)

Hint:

Multiply through by a to clear the fractions, giving 2a² = 4.

Answer 1

The Correct Option is A

Step 1: Set up the equation.
Let the number be a. “Sum of a number and its reciprocal is thrice the difference” gives \(a+\dfrac{1}{a}=3\left(a-\dfrac{1}{a}\right)\).

Step 2: Expand and simplify.
\(a+\dfrac{1}{a}=3a-\dfrac{3}{a} \Rightarrow \dfrac{1}{a}+\dfrac{3}{a}=3a-a \Rightarrow \dfrac{4}{a}=2a\).

Step 3: Solve for a.
\(4=2a^2 \Rightarrow a^2=2 \Rightarrow a=\pm\sqrt{2}\).

Step 4: Final Answer.
The number is \(\pm\sqrt{2}\), so option A is correct.