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

Since the equation mixes $a$ and $\frac{1}{a}$, multiplying through by $a$ early clears the fraction and simplifies the algebra.

1. Start with $a + \frac{1}{a} = 3a - \frac{3}{a}$.
2. Move all $\frac{1}{a}$ terms to one side: $\frac{1}{a} + \frac{3}{a} = 3a - a$, giving $\frac{4}{a} = 2a$.
3. Multiply both sides by $a$: $4 = 2a^2$.
4. Divide by 2: $a^2 = 2$, so $a = \pm\sqrt{2}$.

Checking: with $a=\sqrt{2}$, $\frac{1}{a}=\frac{1}{\sqrt{2}}$, and $a+\frac{1}{a} \approx 2.121$, while $3(a-\frac{1}{a}) \approx 3(0.707) \approx 2.121$, confirming the solution.

So the correct answer is option A, $\pm\sqrt{2}$.