If ` x ^(2) + (1)/( x^(2)) = 66` then the value of ` ( x^(2) - 1 + 2x)/( x)` = ?

Options

A. `pm 8`
B. ` 10 `
C. ` -10`
D. `pm 4`

Answer

To solve the problem, we start with the equation given:

1. **Given Equation**:
\[ x^2 + \frac{1}{x^2} = 66 \]

2. **Let \( n = x - \frac{1}{x} \)**. We know that:
\[ x^2 + \frac{1}{x^2} = \left(x - \frac{1}{x}\right)^2 + 2 = n^2 + 2 \]

3. **Substituting into the equation**:
\[ n^2 + 2 = 66 \]

4. **Solving for \( n^2 \)**:
\[ n^2 = 66 - 2 = 64 \]

5. **Taking the square root**:
\[ n = \sqrt{64} = 8 \quad \text{(We take the positive root since we are looking for a value)} \]

6. **Now we need to find the value of**:
\[ \frac{x^2 - 1 + 2x}{x} \]

7. **Rewriting the expression**:
\[ \frac{x^2 - 1 + 2x}{x} = \frac{x^2}{x} - \frac{1}{x} + \frac{2x}{x} = x - \frac{1}{x} + 2 \]

8. **Substituting the value of \( n \)**:
\[ x - \frac{1}{x} = n = 8 \]
Therefore,
\[ x - \frac{1}{x} + 2 = 8 + 2 = 10 \]

9. **Final Answer**:
\[ \frac{x^2 - 1 + 2x}{x} = 10 \]