If `x=8 + 2sqrt(15)`, then the value of `sqrt(x) + 1/sqrt(x)` is:

Options

A. `2sqrt(3)`
B. `2sqrt(5)`
C. `3/2sqrt(5)+ sqrt(3)/2`
D. `sqrt(5)/2 + 3/2sqrt(3)`

Answer

To solve the problem where \( x = 8 + 2\sqrt{15} \) and we need to find the value of \( \sqrt{x} + \frac{1}{\sqrt{x}} \), we can follow these steps:

### Step 1: Find \( \sqrt{x} \)

Given:
\[ x = 8 + 2\sqrt{15} \]

We can express \( \sqrt{x} \) in a simpler form. We assume:
\[ \sqrt{x} = \sqrt{a} + \sqrt{b} \]

We need to find \( a \) and \( b \) such that:
\[ x = a + b + 2\sqrt{ab} \]

By comparing with \( 8 + 2\sqrt{15} \), we can set:
1. \( a + b = 8 \)
2. \( 2\sqrt{ab} = 2\sqrt{15} \)

From the second equation, we can simplify:
\[ \sqrt{ab} = \sqrt{15} \]
Squaring both sides gives:
\[ ab = 15 \]

### Step 2: Solve for \( a \) and \( b \)

Now we have a system of equations:
1. \( a + b = 8 \)
2. \( ab = 15 \)

We can use these equations to form a quadratic equation:
Let \( t = a \) and \( b = 8 - t \). Then:
\[ t(8 - t) = 15 \]
This simplifies to:
\[ 8t - t^2 = 15 \]
Rearranging gives:
\[ t^2 - 8t + 15 = 0 \]

### Step 3: Factor the quadratic equation

Now we can factor the quadratic:
\[ (t - 3)(t - 5) = 0 \]

Thus, the solutions are:
\[ t = 3 \quad \text{or} \quad t = 5 \]

So, we have:
- \( a = 3 \) and \( b = 5 \) or
- \( a = 5 \) and \( b = 3 \)

### Step 4: Find \( \sqrt{x} \)

Thus:
\[ \sqrt{x} = \sqrt{3} + \sqrt{5} \]

### Step 5: Find \( \frac{1}{\sqrt{x}} \)

Now we need to find \( \frac{1}{\sqrt{x}} \):
\[ \frac{1}{\sqrt{x}} = \frac{1}{\sqrt{3} + \sqrt{5}} \]

To simplify this, we multiply the numerator and denominator by the conjugate:
\[ \frac{\sqrt{3} - \sqrt{5}}{(\sqrt{3} + \sqrt{5})(\sqrt{3} - \sqrt{5})} \]

The denominator becomes:
\[ (\sqrt{3})^2 - (\sqrt{5})^2 = 3 - 5 = -2 \]

Thus:
\[ \frac{1}{\sqrt{x}} = \frac{\sqrt{3} - \sqrt{5}}{-2} = \frac{\sqrt{5} - \sqrt{3}}{2} \]

### Step 6: Combine \( \sqrt{x} \) and \( \frac{1}{\sqrt{x}} \)

Now we can find:
\[ \sqrt{x} + \frac{1}{\sqrt{x}} = \left(\sqrt{3} + \sqrt{5}\right) + \left(\frac{\sqrt{5} - \sqrt{3}}{2}\right) \]

Finding a common denominator (which is 2):
\[ = \frac{2(\sqrt{3} + \sqrt{5})}{2} + \frac{\sqrt{5} - \sqrt{3}}{2} \]
\[ = \frac{2\sqrt{3} + 2\sqrt{5} + \sqrt{5} - \sqrt{3}}{2} \]
\[ = \frac{(2\sqrt{3} - \sqrt{3}) + (2\sqrt{5} + \sqrt{5})}{2} \]
\[ = \frac{\sqrt{3} + 3\sqrt{5}}{2} \]

### Final Result

Thus, the value of \( \sqrt{x} + \frac{1}{\sqrt{x}} \) is:
\[ \frac{\sqrt{3} + 3\sqrt{5}}{2} \]