If x = `8 + 2sqrt(15)` , find `sqrtx+1/(sqrtx)`

Options

A. `2sqrt3`
B. `2sqrt5`
C. `3/2sqrt5+(sqrt3)/2`
D. `(sqrt5)/2+(sqrt3)/(2)sqrt3`

Answer

To solve the problem, we need to find the expression \( \sqrt{x} + \frac{1}{\sqrt{x}} \) given that \( x = 8 + 2\sqrt{15} \).

### Step-by-Step Solution:

1. **Find \( \sqrt{x} \)**:
We start with \( x = 8 + 2\sqrt{15} \). We need to express \( \sqrt{x} \) in a simpler form. We can assume that \( \sqrt{x} = \sqrt{a} + \sqrt{b} \) for some \( a \) and \( b \).

We want:
\[ (\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab} = 8 + 2\sqrt{15} \]
This gives us two equations:
- \( a + b = 8 \)
- \( 2\sqrt{ab} = 2\sqrt{15} \) which simplifies to \( \sqrt{ab} = \sqrt{15} \) or \( ab = 15 \).

2. **Solve the system of equations**:
We have:
\[ a + b = 8 \]
\[ ab = 15 \]
We can treat \( a \) and \( b \) as the roots of the quadratic equation \( t^2 - (a+b)t + ab = 0 \):
\[ t^2 - 8t + 15 = 0 \]

Using the quadratic formula:
\[ t = \frac{8 \pm \sqrt{8^2 - 4 \cdot 1 \cdot 15}}{2 \cdot 1} = \frac{8 \pm \sqrt{64 - 60}}{2} = \frac{8 \pm \sqrt{4}}{2} = \frac{8 \pm 2}{2} \]
This gives us:
\[ t = \frac{10}{2} = 5 \quad \text{and} \quad t = \frac{6}{2} = 3 \]
Thus, \( a = 5 \) and \( b = 3 \) (or vice versa).

3. **Find \( \sqrt{x} \)**:
Therefore, we have:
\[ \sqrt{x} = \sqrt{5} + \sqrt{3} \]

4. **Find \( \frac{1}{\sqrt{x}} \)**:
We need to calculate \( \frac{1}{\sqrt{x}} \):
\[ \frac{1}{\sqrt{x}} = \frac{1}{\sqrt{5} + \sqrt{3}} \]
To simplify, we multiply the numerator and denominator by the conjugate:
\[ \frac{1}{\sqrt{5} + \sqrt{3}} \cdot \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}} = \frac{\sqrt{5} - \sqrt{3}}{(\sqrt{5})^2 - (\sqrt{3})^2} = \frac{\sqrt{5} - \sqrt{3}}{5 - 3} = \frac{\sqrt{5} - \sqrt{3}}{2} \]

5. **Combine \( \sqrt{x} \) and \( \frac{1}{\sqrt{x}} \)**:
Now we can find \( \sqrt{x} + \frac{1}{\sqrt{x}} \):
\[ \sqrt{x} + \frac{1}{\sqrt{x}} = \left(\sqrt{5} + \sqrt{3}\right) + \left(\frac{\sqrt{5} - \sqrt{3}}{2}\right) \]
To combine these, we need a common denominator:
\[ = \frac{2(\sqrt{5} + \sqrt{3}) + (\sqrt{5} - \sqrt{3})}{2} = \frac{2\sqrt{5} + 2\sqrt{3} + \sqrt{5} - \sqrt{3}}{2} = \frac{(2\sqrt{5} + \sqrt{5}) + (2\sqrt{3} - \sqrt{3})}{2} \]
\[ = \frac{3\sqrt{5} + \sqrt{3}}{2} \]

### Final Answer:
Thus, the final result is:
\[ \sqrt{x} + \frac{1}{\sqrt{x}} = \frac{3\sqrt{5} + \sqrt{3}}{2} \]