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

Options

A. `26`
B. `23`
C. `30`
D. `22`

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:
\[ \left(x + \frac{1}{x}\right)^2 = 5^2 \]

### Step 2: Apply the square of a binomial formula
Using the identity \( (a + b)^2 = a^2 + b^2 + 2ab \), we can expand the left side:
\[ x^2 + 2\left(x \cdot \frac{1}{x}\right) + \frac{1}{x^2} = 25 \]
Since \( x \cdot \frac{1}{x} = 1 \), we can simplify this to:
\[ x^2 + 2 + \frac{1}{x^2} = 25 \]

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

### Step 4: Simplify the result
Now, we can simplify the right side:
\[ x^2 + \frac{1}{x^2} = 23 \]

Thus, the value of \( x^2 + \frac{1}{x^2} \) is \( 23 \).

### Final Answer
\[ \boxed{23} \]