The sum of a number and its reciprocal is equal to twice the difference of the number and its reciprocal. The number is equal to :

Question

Accepted Answer

To solve the problem, we need to find a number $ x $ such that the sum of the number and its reciprocal is equal to twice the difference of the number and its reciprocal. Mathematically, this can be expressed as:

$$ x + \frac{1}{x} = 2 \left( x - \frac{1}{x} \right) $$

Let's simplify and solve this equation step by step:

1. **Expand the right side of the equation:**

$$ x + \frac{1}{x} = 2x - 2 \cdot \frac{1}{x} $$

2. **Rearrange the equation:**

$$ x + \frac{1}{x} = 2x - \frac{2}{x} $$

3. **Bring all terms to one side:**

$$ x + \frac{1}{x} - 2x + \frac{2}{x} = 0 $$

$$ -x + \frac{3}{x} = 0 $$

4. **Multiply through by $ x $ to eliminate the fraction:**

$$ -x^2 + 3 = 0 $$

5. **Solve for $ x^2 $:**

$$ x^2 = 3 $$

6. **Take the square root of both sides:**

$$ x = \pm \sqrt{3} $$

Thus, the number that satisfies the given condition is $ \pm \sqrt{3} $, which corresponds to Option 3.

### Explanation of Why Other Options are Incorrect:

1. **Option 1: 2**

If $ x = 2 $, then the reciprocal is $ \frac{1}{2} $. The sum $ x + \frac{1}{x} = 2 + \frac{1}{2} = 2.5 $. The difference $ x - \frac{1}{x} = 2 - \frac{1}{2} = 1.5 $. Twice the difference is $ 2 \times 1.5 = 3 $. Since 2.5 is not equal to 3, Option 1 is incorrect.

2. **Option 2: 3**

If $ x = 3 $, then the reciprocal is $ \frac{1}{3} $. The sum $ x + \frac{1}{x} = 3 + \frac{1}{3} = 3.333\ldots $. The difference $ x - \frac{1}{x} = 3 - \frac{1}{3} = 2.666\ldots $. Twice the difference is $ 2 \times 2.666\ldots = 5.333\ldots $. Since 3.333\ldots is not equal to 5.333\ldots, Option 2 is incorrect.

3. **Option 4: $\pm \sqrt{2}$**

If $ x = \sqrt{2} $, then the reciprocal is $ \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} $. The sum $ x + \frac{1}{x} = \sqrt{2} + \frac{\sqrt{2}}{2} = \frac{3\sqrt{2}}{2} $. The difference $ x - \frac{1}{x} = \sqrt{2} - \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2} $. Twice the difference is $ 2 \times \frac{\sqrt{2}}{2} = \sqrt{2} $. Since $\frac{3\sqrt{2}}{2}$ is not equal to $\sqrt{2}$, Option 4 is incorrect.

Therefore, the correct answer is indeed Option 3: $ \pm \sqrt{3} $.