If `x = (3+ sqrt7)/(2)` , then the value of `4x^2 + 1/x^2` is ______ .

Options

A. 18
B. `3-2sqrt7`
C. 32
D. `1/(3+2sqrt7)`

Answer

To find the value of \( 4x^2 + \frac{1}{x^2} \) given \( x = \frac{3 + \sqrt{7}}{2} \), we can follow these steps:

### Step 1: Calculate \( 2x \)
First, we can express \( 2x \):
\[ 2x = 3 + \sqrt{7} \]

### Step 2: Square \( 2x \)
Next, we square both sides to find \( 4x^2 \):
\[ (2x)^2 = (3 + \sqrt{7})^2 \]
Expanding the right side using the formula \( (a + b)^2 = a^2 + b^2 + 2ab \):
\[ 4x^2 = 3^2 + (\sqrt{7})^2 + 2 \cdot 3 \cdot \sqrt{7} \]
Calculating each term:
\[ 4x^2 = 9 + 7 + 6\sqrt{7} \]
Combining the terms:
\[ 4x^2 = 16 + 6\sqrt{7} \]

### Step 3: Calculate \( \frac{1}{x} \)
Now, we need to find \( \frac{1}{x} \):
\[ \frac{1}{x} = \frac{2}{3 + \sqrt{7}} \]
To rationalize the denominator, multiply the numerator and denominator by the conjugate \( 3 - \sqrt{7} \):
\[ \frac{1}{x} = \frac{2(3 - \sqrt{7})}{(3 + \sqrt{7})(3 - \sqrt{7})} \]
Calculating the denominator:
\[ (3 + \sqrt{7})(3 - \sqrt{7}) = 9 - 7 = 2 \]
Thus,
\[ \frac{1}{x} = \frac{2(3 - \sqrt{7})}{2} = 3 - \sqrt{7} \]

### Step 4: Calculate \( \frac{1}{x^2} \)
Now, we square \( \frac{1}{x} \):
\[ \frac{1}{x^2} = (3 - \sqrt{7})^2 \]
Expanding this:
\[ \frac{1}{x^2} = 3^2 + (\sqrt{7})^2 - 2 \cdot 3 \cdot \sqrt{7} \]
Calculating each term:
\[ \frac{1}{x^2} = 9 + 7 - 6\sqrt{7} \]
Combining the terms:
\[ \frac{1}{x^2} = 16 - 6\sqrt{7} \]

### Step 5: Combine \( 4x^2 \) and \( \frac{1}{x^2} \)
Now, we can find \( 4x^2 + \frac{1}{x^2} \):
\[ 4x^2 + \frac{1}{x^2} = (16 + 6\sqrt{7}) + (16 - 6\sqrt{7}) \]
The \( 6\sqrt{7} \) terms cancel out:
\[ 4x^2 + \frac{1}{x^2} = 16 + 16 = 32 \]

### Final Answer
Thus, the value of \( 4x^2 + \frac{1}{x^2} \) is:
\[ \boxed{32} \]