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

Options

A. `+- sqrt(2)`
B. `+- 1/(sqrt2)`
C. `+- sqrt(3)`
D. `+- 1/(sqrt3)`

Answer

To solve the problem, we need to find a number \( A \) such that the sum of the number and its reciprocal is thrice the difference of the number and its reciprocal. Let's break this down step by step.

### Step 1: Set up the equation
Let the number be \( A \). The reciprocal of \( A \) is \( \frac{1}{A} \). According to the problem, we have:

\[ A + \frac{1}{A} = 3 \left( A - \frac{1}{A} \right) \]

### Step 2: Simplify the equation
Now, we will simplify both sides of the equation. The left side remains as is:

\[ A + \frac{1}{A} \]

The right side can be expanded:

\[ 3 \left( A - \frac{1}{A} \right) = 3A - \frac{3}{A} \]

So we can rewrite the equation as:

\[ A + \frac{1}{A} = 3A - \frac{3}{A} \]

### Step 3: Eliminate the fractions
To eliminate the fractions, multiply every term by \( A \) (assuming \( A \neq 0 \)):

\[ A^2 + 1 = 3A^2 - 3 \]

### Step 4: Rearrange the equation
Now, let's move all terms to one side of the equation:

\[ A^2 + 1 - 3A^2 + 3 = 0 \]

This simplifies to:

\[ -2A^2 + 4 = 0 \]

### Step 5: Solve for \( A^2 \)
Rearranging gives us:

\[ 2A^2 = 4 \]

Dividing both sides by 2:

\[ A^2 = 2 \]

### Step 6: Find \( A \)
Taking the square root of both sides, we get:

\[ A = \sqrt{2} \quad \text{or} \quad A = -\sqrt{2} \]

### Final Answer
The number is \( \sqrt{2} \) or \( -\sqrt{2} \).

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