If `x ^(2) + 2 sqrt10 x + 1=0,` then What is the value of `x - (1)/(x)` ?

Options

A. 4
B. 6
C. 3
D. 5

Answer

To solve the equation \( x^2 + 2\sqrt{10}x + 1 = 0 \) and find the value of \( x - \frac{1}{x} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ x^2 + 2\sqrt{10}x + 1 = 0 \]

### Step 2: Divide the entire equation by \( x \)
To simplify the equation, we divide every term by \( x \) (assuming \( x \neq 0 \)):
\[ \frac{x^2}{x} + \frac{2\sqrt{10}x}{x} + \frac{1}{x} = 0 \]
This simplifies to:
\[ x + 2\sqrt{10} + \frac{1}{x} = 0 \]

### Step 3: Rearrange to find \( x + \frac{1}{x} \)
From the equation above, we can isolate \( x + \frac{1}{x} \):
\[ x + \frac{1}{x} = -2\sqrt{10} \]

### Step 4: Use the identity for \( x - \frac{1}{x} \)
We need to find \( x - \frac{1}{x} \). We can use the identity:
\[ \left( x - \frac{1}{x} \right)^2 = \left( x + \frac{1}{x} \right)^2 - 4 \]

### Step 5: Calculate \( \left( x + \frac{1}{x} \right)^2 \)
First, we calculate \( \left( x + \frac{1}{x} \right)^2 \):
\[ \left( -2\sqrt{10} \right)^2 = 4 \cdot 10 = 40 \]

### Step 6: Substitute into the identity
Now we substitute this into the identity:
\[ \left( x - \frac{1}{x} \right)^2 = 40 - 4 = 36 \]

### Step 7: Take the square root
To find \( x - \frac{1}{x} \), we take the square root:
\[ x - \frac{1}{x} = \sqrt{36} = 6 \quad \text{or} \quad x - \frac{1}{x} = -6 \]

Since we are looking for the positive value, we conclude:
\[ x - \frac{1}{x} = 6 \]

### Final Answer
Thus, the value of \( x - \frac{1}{x} \) is:
\[ \boxed{6} \]