If `(x - (1)/(x)) = 5`, find the value of `(x^(2) + (1)/(x^(2)))`

Options

A. 34
B. 42
C. 27
D. 63

Answer

To solve the equation \( x - \frac{1}{x} = 5 \) and find the value of \( x^2 + \frac{1}{x^2} \), we can follow these steps:

### Step 1: Square both sides of the equation
Starting with the equation:
\[ x - \frac{1}{x} = 5 \]
We square both sides:
\[ \left( x - \frac{1}{x} \right)^2 = 5^2 \]

### Step 2: Expand the left-hand side
Using the identity \( (a - b)^2 = a^2 - 2ab + b^2 \):
\[ x^2 - 2 \cdot x \cdot \frac{1}{x} + \left( \frac{1}{x} \right)^2 = 25 \]
This simplifies to:
\[ x^2 - 2 + \frac{1}{x^2} = 25 \]

### Step 3: Combine like terms
Rearranging the equation gives:
\[ x^2 + \frac{1}{x^2} - 2 = 25 \]

### Step 4: Isolate \( x^2 + \frac{1}{x^2} \)
Adding 2 to both sides:
\[ x^2 + \frac{1}{x^2} = 25 + 2 \]

### Step 5: Calculate the final result
Thus, we find:
\[ x^2 + \frac{1}{x^2} = 27 \]

### Final Answer
The value of \( x^2 + \frac{1}{x^2} \) is \( \boxed{27} \).