If `x - (1)/(x) = 4` , then `(x + (1)/(x))` is equal to

Options

A. `5 sqrt2`
B. `2 sqrt5`
C. `4 sqrt2`
D. `4 sqrt5`

Answer

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

### Step 1: Square both sides of the equation
Starting with the given equation:
\[ x - \frac{1}{x} = 4 \]
We square both sides:
\[ \left(x - \frac{1}{x}\right)^2 = 4^2 \]
This simplifies to:
\[ x^2 - 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} = 16 \]
Which further simplifies to:
\[ x^2 - 2 + \frac{1}{x^2} = 16 \]

### Step 2: Rearrange the equation
Now, we can rearrange the equation:
\[ x^2 + \frac{1}{x^2} - 2 = 16 \]
Adding 2 to both sides gives us:
\[ x^2 + \frac{1}{x^2} = 16 + 2 \]
Thus:
\[ x^2 + \frac{1}{x^2} = 18 \]

### Step 3: Use the identity to find \( x + \frac{1}{x} \)
We know the identity:
\[ \left(x + \frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2} \]
We can express \( x^2 + \frac{1}{x^2} \) in terms of \( x + \frac{1}{x} \):
\[ \left(x + \frac{1}{x}\right)^2 = x^2 + \frac{1}{x^2} + 2 \]
Substituting the value we found:
\[ \left(x + \frac{1}{x}\right)^2 = 18 + 2 \]
This simplifies to:
\[ \left(x + \frac{1}{x}\right)^2 = 20 \]

### Step 4: Take the square root
Taking the square root of both sides gives us:
\[ x + \frac{1}{x} = \sqrt{20} \]
This can be simplified to:
\[ x + \frac{1}{x} = 2\sqrt{5} \]

### Final Answer
Thus, the value of \( x + \frac{1}{x} \) is:
\[ \boxed{2\sqrt{5}} \]