`x=(4)/(2sqrt(3)+3sqrt(2))` then find the value of `(x + (1)/(x))`

Options

A. a. `((10sqrt(3)+15sqrt(2)))/(12)`
B. b. `((10sqrt(3)-15sqrt2))/(12)`
C. c. `(-(10sqrt(3)-33sqrt(2)))/(12)`
D. d. `((10sqrt(3)+33sqrt(2)))/(12)`

Answer

To solve the problem \( x = \frac{4}{2\sqrt{3} + 3\sqrt{2}} \) and find the value of \( x + \frac{1}{x} \), we can follow these steps:

### Step 1: Find \( \frac{1}{x} \)
To find \( \frac{1}{x} \), we take the reciprocal of \( x \):
\[ \frac{1}{x} = \frac{2\sqrt{3} + 3\sqrt{2}}{4} \]

### Step 2: Rationalize \( x \)
To simplify \( x \), we can rationalize the denominator:
\[ x = \frac{4}{2\sqrt{3} + 3\sqrt{2}} \cdot \frac{2\sqrt{3} - 3\sqrt{2}}{2\sqrt{3} - 3\sqrt{2}} = \frac{4(2\sqrt{3} - 3\sqrt{2})}{(2\sqrt{3})^2 - (3\sqrt{2})^2} \]
Calculating the denominator:
\[ (2\sqrt{3})^2 = 4 \cdot 3 = 12 \quad \text{and} \quad (3\sqrt{2})^2 = 9 \cdot 2 = 18 \]
Thus,
\[ (2\sqrt{3})^2 - (3\sqrt{2})^2 = 12 - 18 = -6 \]
Now substituting back:
\[ x = \frac{4(2\sqrt{3} - 3\sqrt{2})}{-6} = -\frac{2(2\sqrt{3} - 3\sqrt{2})}{3} \]

### Step 3: Simplify \( x \)
This simplifies to:
\[ x = -\frac{4\sqrt{3}}{3} + 2\sqrt{2} \]

### Step 4: Calculate \( x + \frac{1}{x} \)
Now we need to find \( x + \frac{1}{x} \):
\[ x + \frac{1}{x} = \left(-\frac{4\sqrt{3}}{3} + 2\sqrt{2}\right) + \left(\frac{2\sqrt{3} + 3\sqrt{2}}{4}\right) \]

### Step 5: Find a common denominator
The common denominator for the terms is 12:
\[ x + \frac{1}{x} = \left(-\frac{16\sqrt{3}}{12} + \frac{24\sqrt{2}}{12}\right) + \left(\frac{6\sqrt{3}}{12} + \frac{9\sqrt{2}}{12}\right) \]

### Step 6: Combine the fractions
Combining the fractions:
\[ x + \frac{1}{x} = \frac{-16\sqrt{3} + 6\sqrt{3} + 24\sqrt{2} + 9\sqrt{2}}{12} = \frac{-10\sqrt{3} + 33\sqrt{2}}{12} \]

### Step 7: Final result
Thus, the final value of \( x + \frac{1}{x} \) is:
\[ x + \frac{1}{x} = \frac{33\sqrt{2} - 10\sqrt{3}}{12} \]