If `x^(2)- 2sqrt(10)x+1=0` then what is the value of `x- (1)/(x)?`

Options

A. A)4
B. B)6
C. C)3
D. 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: Solve the quadratic equation
We start with the quadratic equation:
\[ x^2 - 2\sqrt{10}x + 1 = 0 \]
We can use the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -2\sqrt{10} \), and \( c = 1 \).

### Step 2: Calculate the discriminant
First, we calculate the discriminant \( b^2 - 4ac \):
\[ b^2 = (-2\sqrt{10})^2 = 4 \cdot 10 = 40 \]
\[ 4ac = 4 \cdot 1 \cdot 1 = 4 \]
Thus, the discriminant is:
\[ b^2 - 4ac = 40 - 4 = 36 \]

### Step 3: Find the roots
Now substituting back into the quadratic formula:
\[ x = \frac{2\sqrt{10} \pm \sqrt{36}}{2 \cdot 1} = \frac{2\sqrt{10} \pm 6}{2} \]
This simplifies to:
\[ x = \sqrt{10} + 3 \quad \text{or} \quad x = \sqrt{10} - 3 \]

### Step 4: Calculate \( x + \frac{1}{x} \)
Next, we need to find \( x + \frac{1}{x} \). We can use the relationship:
\[ x + \frac{1}{x} = 2\sqrt{10} \]
This is derived from the original equation by dividing through by \( x \):
\[ x - 2\sqrt{10} + \frac{1}{x} = 0 \implies x + \frac{1}{x} = 2\sqrt{10} \]

### Step 5: Find \( x - \frac{1}{x} \)
Using the identity:
\[ x - \frac{1}{x} = \sqrt{(x + \frac{1}{x})^2 - 4} \]
Substituting \( x + \frac{1}{x} = 2\sqrt{10} \):
\[ x - \frac{1}{x} = \sqrt{(2\sqrt{10})^2 - 4} = \sqrt{40 - 4} = \sqrt{36} = 6 \]

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