If `x=5+2sqrt(6)`, then what is the value of `(sqrt(x)+(1)/(sqrt(x)))` ?

Options

A. `2sqrt(3)`
B. `3sqrt(2)`
C. `2sqrt(6)`
D. `6sqrt(2)`

Answer

To find the value of \( \sqrt{x} + \frac{1}{\sqrt{x}} \) where \( x = 5 + 2\sqrt{6} \), we can follow these steps:

### Step 1: Find \( \sqrt{x} \)
We start with \( x = 5 + 2\sqrt{6} \). We need to express \( \sqrt{x} \) in a simpler form. We can assume that \( \sqrt{x} \) can be expressed as \( \sqrt{a} + \sqrt{b} \).

### Step 2: Set up the equation
Assuming \( \sqrt{x} = \sqrt{a} + \sqrt{b} \), we square both sides:
\[ x = (\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab} \]
From this, we can equate the rational and irrational parts:
1. \( a + b = 5 \)
2. \( 2\sqrt{ab} = 2\sqrt{6} \)

### Step 3: Solve for \( ab \)
From the second equation, we can simplify:
\[ \sqrt{ab} = \sqrt{6} \implies ab = 6 \]

### Step 4: Solve the system of equations
Now we have the system of equations:
1. \( a + b = 5 \)
2. \( ab = 6 \)

We can treat this as a quadratic equation:
Let \( t \) be \( a \) and \( b \). Then:
\[ t^2 - (a+b)t + ab = 0 \implies t^2 - 5t + 6 = 0 \]

### Step 5: Factor the quadratic
Factoring the quadratic:
\[ (t - 2)(t - 3) = 0 \]
Thus, \( t = 2 \) or \( t = 3 \). Therefore, \( a = 2 \) and \( b = 3 \) (or vice versa).

### Step 6: Find \( \sqrt{x} \)
Now we can express \( \sqrt{x} \):
\[ \sqrt{x} = \sqrt{2} + \sqrt{3} \]

### Step 7: Calculate \( \frac{1}{\sqrt{x}} \)
To find \( \frac{1}{\sqrt{x}} \):
\[ \frac{1}{\sqrt{x}} = \frac{1}{\sqrt{2} + \sqrt{3}} \cdot \frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} - \sqrt{3}} = \frac{\sqrt{2} - \sqrt{3}}{2 - 3} = -(\sqrt{2} - \sqrt{3}) = \sqrt{3} - \sqrt{2} \]

### Step 8: Combine the results
Now we can combine \( \sqrt{x} \) and \( \frac{1}{\sqrt{x}} \):
\[ \sqrt{x} + \frac{1}{\sqrt{x}} = (\sqrt{2} + \sqrt{3}) + (\sqrt{3} - \sqrt{2}) = 2\sqrt{3} \]

### Final Answer
Thus, the value of \( \sqrt{x} + \frac{1}{\sqrt{x}} \) is:
\[ \boxed{2\sqrt{3}} \]