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

Answer

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

### Step 1: Rewrite the equation
We start with the equation:
\[ x^2 = 11 + 2\sqrt{30} \]

### Step 2: Express \( x^2 \) in a different form
We can express \( 11 + 2\sqrt{30} \) in a way that resembles a perfect square. Notice that:
\[ 11 + 2\sqrt{30} = 6 + 5 + 2\sqrt{5 \cdot 6} \]
This can be rewritten as:
\[ x^2 = (\sqrt{6} + \sqrt{5})^2 \]

### Step 3: Take the square root
Taking the square root of both sides gives us:
\[ x = \pm (\sqrt{6} + \sqrt{5}) \]

### Step 4: Find \( x + \frac{1}{x} \)
Now, we need to find \( x + \frac{1}{x} \). We can use the positive value of \( x \):
\[ x + \frac{1}{x} = \sqrt{6} + \sqrt{5} + \frac{1}{\sqrt{6} + \sqrt{5}} \]

### Step 5: Rationalize \( \frac{1}{\sqrt{6} + \sqrt{5}} \)
To simplify \( \frac{1}{\sqrt{6} + \sqrt{5}} \), we multiply the numerator and denominator by \( \sqrt{6} - \sqrt{5} \):
\[ \frac{1}{\sqrt{6} + \sqrt{5}} \cdot \frac{\sqrt{6} - \sqrt{5}}{\sqrt{6} - \sqrt{5}} = \frac{\sqrt{6} - \sqrt{5}}{(\sqrt{6})^2 - (\sqrt{5})^2} \]
Calculating the denominator:
\[ (\sqrt{6})^2 - (\sqrt{5})^2 = 6 - 5 = 1 \]
Thus,
\[ \frac{1}{\sqrt{6} + \sqrt{5}} = \sqrt{6} - \sqrt{5} \]

### Step 6: Combine the terms
Now, we can combine the terms:
\[ x + \frac{1}{x} = \sqrt{6} + \sqrt{5} + (\sqrt{6} - \sqrt{5}) = 2\sqrt{6} \]

### Final Answer
Thus, the final result is:
\[ x + \frac{1}{x} = 2\sqrt{6} \]

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