If `(x-1/x)=4` then the value of `(x^2+1/x^2)` is

Options

A. 16
B. 18
C. `18sqrt3`
D. none of these

Answer

To solve the problem where \( x - \frac{1}{x} = 4 \) and we need to 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:
\[ x - \frac{1}{x} = 4 \]
Now, we square both sides:
\[ \left(x - \frac{1}{x}\right)^2 = 4^2 \]
This simplifies to:
\[ x^2 - 2\left(x \cdot \frac{1}{x}\right) + \left(\frac{1}{x}\right)^2 = 16 \]

### Step 2: Simplify the left-hand side
Using the identity \( a^2 - 2ab + b^2 = (a-b)^2 \), we can rewrite the left-hand side:
\[ x^2 - 2 + \frac{1}{x^2} = 16 \]
Here, \( 2\left(x \cdot \frac{1}{x}\right) = 2 \).

### Step 3: Rearranging the equation
Now we rearrange the equation to isolate \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} - 2 = 16 \]
Adding 2 to both sides gives:
\[ x^2 + \frac{1}{x^2} = 16 + 2 \]

### Step 4: Final calculation
This simplifies to:
\[ x^2 + \frac{1}{x^2} = 18 \]

### Conclusion
Thus, the value of \( x^2 + \frac{1}{x^2} \) is \( \boxed{18} \).