If ` x- (1)/ (x ) = 4`, find the value of :
` x^(2) + (1)/( x^(2))`

Answer

To solve the equation \( x - \frac{1}{x} = 4 \) and find the value of \( x^2 + \frac{1}{x^2} \), we can follow these steps:

### Step 1: Start with the given equation
We have:
\[ x - \frac{1}{x} = 4 \]

### Step 2: Square both sides
Squaring both sides gives us:
\[ \left(x - \frac{1}{x}\right)^2 = 4^2 \]
This simplifies to:
\[ x^2 - 2\cdot x\cdot \frac{1}{x} + \frac{1}{x^2} = 16 \]

### Step 3: Simplify the equation
The term \( -2\cdot x\cdot \frac{1}{x} \) simplifies to \( -2 \), so we have:
\[ x^2 - 2 + \frac{1}{x^2} = 16 \]

### Step 4: Rearrange the equation
Now, we can rearrange the equation to isolate \( x^2 + \frac{1}{x^2} \):
\[ x^2 + \frac{1}{x^2} = 16 + 2 \]

### Step 5: Calculate the final value
Adding the numbers gives us:
\[ x^2 + \frac{1}{x^2} = 18 \]

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

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