If `x=2+sqrt(3)`, then the value of `(x^(2)-x+1)/(x^(2)+x+1)` is :

Options

A. `(2)/(3)`
B. `(3)/(4)`
C. `(4)/(5)`
D. `(3)/(5)`

Answer

To solve the problem, we need to find the value of the expression \((x^2 - x + 1) / (x^2 + x + 1)\) given that \(x = 2 + \sqrt{3}\).

### Step-by-Step Solution:

1. **Calculate \(x^2\)**:
\[ x = 2 + \sqrt{3} \]
Squaring \(x\):
\[ x^2 = (2 + \sqrt{3})^2 = 2^2 + 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \]

**Hint**: Remember to use the formula \((a + b)^2 = a^2 + 2ab + b^2\).

2. **Calculate \(x^2 - x + 1\)**:
\[ x^2 - x + 1 = (7 + 4\sqrt{3}) - (2 + \sqrt{3}) + 1 \]
Simplifying this:
\[ = 7 + 4\sqrt{3} - 2 - \sqrt{3} + 1 = (7 - 2 + 1) + (4\sqrt{3} - \sqrt{3}) = 6 + 3\sqrt{3} \]

**Hint**: Combine like terms carefully.

3. **Calculate \(x^2 + x + 1\)**:
\[ x^2 + x + 1 = (7 + 4\sqrt{3}) + (2 + \sqrt{3}) + 1 \]
Simplifying this:
\[ = 7 + 4\sqrt{3} + 2 + \sqrt{3} + 1 = (7 + 2 + 1) + (4\sqrt{3} + \sqrt{3}) = 10 + 5\sqrt{3} \]

**Hint**: Again, combine like terms carefully.

4. **Form the final expression**:
Now we substitute back into the original expression:
\[ \frac{x^2 - x + 1}{x^2 + x + 1} = \frac{6 + 3\sqrt{3}}{10 + 5\sqrt{3}} \]

**Hint**: Ensure you write the fraction correctly.

5. **Simplify the fraction**:
To simplify, we can factor out the common terms:
\[ = \frac{3(2 + \sqrt{3})}{5(2 + \sqrt{3})} = \frac{3}{5} \]

**Hint**: Look for common factors in the numerator and denominator.

### Final Answer:
The value of \(\frac{x^2 - x + 1}{x^2 + x + 1}\) is \(\frac{3}{5}\).