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

Answer

To solve the equation \( x^2 = 11 + 2\sqrt{30} \), we will follow these steps:

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

### Step 2: Express 11 in a different form
We can express 11 as a sum of two squares. We can rewrite it as:
\[ 11 = 6 + 5 \]
Now we can rewrite the right-hand side:
\[ x^2 = (6 + 5) + 2\sqrt{30} \]

### Step 3: Recognize the perfect square
Notice that \( 6 = \sqrt{6}^2 \) and \( 5 = \sqrt{5}^2 \). We can now express the right-hand side as:
\[ x^2 = \sqrt{6}^2 + \sqrt{5}^2 + 2(\sqrt{5})(\sqrt{6}) \]
This is in the form of \( a^2 + b^2 + 2ab \), which can be factored as:
\[ x^2 = (\sqrt{6} + \sqrt{5})^2 \]

### Step 4: Take the square root of both sides
Now, we can take the square root of both sides:
\[ x = \pm (\sqrt{6} + \sqrt{5}) \]

### Final Answer
Thus, the value of \( x \) is:
\[ x = \sqrt{6} + \sqrt{5} \quad \text{or} \quad x = -(\sqrt{6} + \sqrt{5}) \]

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