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

Options

A. 25
B. 27
C. 23
D. `25 (1)/(25)`

Answer

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

### Step 1: Square both sides of the equation
Starting with the given equation:
\[ x - \frac{1}{x} = 5 \]
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 left side
The middle term simplifies as follows:
\[ -2 \cdot x \cdot \frac{1}{x} = -2 \]
So we can rewrite the equation as:
\[ x^2 - 2 + \frac{1}{x^2} = 25 \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to isolate \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = 25 + 2 \]
This simplifies to:
\[ x^2 + \frac{1}{x^2} = 27 \]

### Final Answer
Thus, the value of \( x^2 + \frac{1}{x^2} \) is:
\[ \boxed{27} \]

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