If ` x - (1)/(x) = 5` then ` x^(2) + (1)/( x^(2))` is

Options

A. 5
B. 25
C. 27
D. 23

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
We start with the equation:
\[ x - \frac{1}{x} = 5 \]
Now, we square both sides:
\[ \left( x - \frac{1}{x} \right)^2 = 5^2 \]
This simplifies to:
\[ x^2 - 2 \cdot x \cdot \frac{1}{x} + \left( \frac{1}{x} \right)^2 = 25 \]

### Step 2: Simplify the equation
The term \( -2 \cdot x \cdot \frac{1}{x} \) simplifies to \(-2\):
\[ x^2 - 2 + \frac{1}{x^2} = 25 \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to isolate \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = 25 + 2 \]

### Step 4: Calculate the final value
Adding the numbers gives us:
\[ x^2 + \frac{1}{x^2} = 27 \]

Thus, the value of \( x^2 + \frac{1}{x^2} \) is \( \boxed{27} \).

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