If `x = 2 + sqrt3,` 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 \(\frac{x^2 - x + 1}{x^2 + x + 1}\) given that \(x = 2 + \sqrt{3}\).

### Step-by-Step Solution:

1. **Substituting the value of \(x\)**:
We start by substituting \(x = 2 + \sqrt{3}\) into the expression:
\[ \frac{(2 + \sqrt{3})^2 - (2 + \sqrt{3}) + 1}{(2 + \sqrt{3})^2 + (2 + \sqrt{3}) + 1} \]

2. **Calculating \(x^2\)**:
First, we calculate \(x^2\):
\[ x^2 = (2 + \sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \]

3. **Substituting \(x^2\) into the expression**:
Now we substitute \(x^2\) back into the expression:
\[ \frac{(7 + 4\sqrt{3}) - (2 + \sqrt{3}) + 1}{(7 + 4\sqrt{3}) + (2 + \sqrt{3}) + 1} \]

4. **Simplifying the numerator**:
Simplifying the numerator:
\[ 7 + 4\sqrt{3} - 2 - \sqrt{3} + 1 = 6 + 3\sqrt{3} \]

5. **Simplifying the denominator**:
Now simplifying the denominator:
\[ 7 + 4\sqrt{3} + 2 + \sqrt{3} + 1 = 10 + 5\sqrt{3} \]

6. **Final expression**:
We now have:
\[ \frac{6 + 3\sqrt{3}}{10 + 5\sqrt{3}} \]

7. **Rationalizing the denominator**:
To simplify further, we multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{(6 + 3\sqrt{3})(10 - 5\sqrt{3})}{(10 + 5\sqrt{3})(10 - 5\sqrt{3})} \]

8. **Calculating the denominator**:
The denominator becomes:
\[ 10^2 - (5\sqrt{3})^2 = 100 - 75 = 25 \]

9. **Calculating the numerator**:
The numerator expands as follows:
\[ (6 \cdot 10) + (6 \cdot -5\sqrt{3}) + (3\sqrt{3} \cdot 10) + (3\sqrt{3} \cdot -5\sqrt{3}) = 60 - 30\sqrt{3} + 30\sqrt{3} - 45 = 15 \]

10. **Final result**:
Thus, we have:
\[ \frac{15}{25} = \frac{3}{5} \]

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