If ` x ^(2) = 11 + 2 sqrt(30 ) , ` find :
` x - (1)/(x)`

Answer

To solve the problem where \( x^2 = 11 + 2\sqrt{30} \) and we need to find \( x - \frac{1}{x} \), we can follow these steps:

### Step 1: Find \( x \)
We start with the equation:
\[ x^2 = 11 + 2\sqrt{30} \]
To find \( x \), we take the square root of both sides:
\[ x = \sqrt{11 + 2\sqrt{30}} \]

### Step 2: Simplify \( \sqrt{11 + 2\sqrt{30}} \)
Next, we will express \( 11 \) as \( 6 + 5 \) to use the identity \( (a + b)^2 = a^2 + b^2 + 2ab \):
\[ x = \sqrt{6 + 5 + 2\sqrt{30}} \]
This can be rewritten as:
\[ x = \sqrt{(\sqrt{6} + \sqrt{5})^2} \]
Thus, we have:
\[ x = \sqrt{6} + \sqrt{5} \]

### Step 3: Find \( \frac{1}{x} \)
Now we need to find \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{1}{\sqrt{6} + \sqrt{5}} \]
To rationalize the denominator, we multiply the numerator and the denominator by \( \sqrt{6} - \sqrt{5} \):
\[ \frac{1}{x} = \frac{\sqrt{6} - \sqrt{5}}{(\sqrt{6} + \sqrt{5})(\sqrt{6} - \sqrt{5})} \]
Calculating the denominator using the difference of squares:
\[ (\sqrt{6})^2 - (\sqrt{5})^2 = 6 - 5 = 1 \]
So, we have:
\[ \frac{1}{x} = \sqrt{6} - \sqrt{5} \]

### Step 4: Calculate \( x - \frac{1}{x} \)
Now we can find \( x - \frac{1}{x} \):
\[ x - \frac{1}{x} = (\sqrt{6} + \sqrt{5}) - (\sqrt{6} - \sqrt{5}) \]
Simplifying this gives:
\[ x - \frac{1}{x} = \sqrt{6} + \sqrt{5} - \sqrt{6} + \sqrt{5} = 2\sqrt{5} \]

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