Solve the equation `x^(2)+x+(1)/(sqrt(3))=0`

Explanation

To solve the equation \( x^2 + x + \frac{1}{\sqrt{3}} = 0 \), we will use the quadratic formula.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given equation is in the standard form \( ax^2 + bx + c = 0 \).
Here, \( a = 1 \), \( b = 1 \), and \( c = \frac{1}{\sqrt{3}} \).

2. **Apply the quadratic formula**:
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot \frac{1}{\sqrt{3}}}}{2 \cdot 1} \]

3. **Calculate the discriminant**:
First, calculate \( b^2 - 4ac \):
\[ b^2 = 1^2 = 1 \]
\[ 4ac = 4 \cdot 1 \cdot \frac{1}{\sqrt{3}} = \frac{4}{\sqrt{3}} \]
Now, substitute these values into the discriminant:
\[ 1 - \frac{4}{\sqrt{3}} = \frac{\sqrt{3}}{\sqrt{3}} - \frac{4}{\sqrt{3}} = \frac{\sqrt{3} - 4}{\sqrt{3}} \]

4. **Substitute back into the quadratic formula**:
Now, substituting the discriminant back into the formula:
\[ x = \frac{-1 \pm \sqrt{\frac{\sqrt{3} - 4}{\sqrt{3}}}}{2} \]

5. **Simplify the square root**:
The square root can be simplified:
\[ x = \frac{-1 \pm \frac{\sqrt{\sqrt{3} - 4}}{\sqrt{\sqrt{3}}}}{2} \]
This gives:
\[ x = \frac{-1 \pm \sqrt{\sqrt{3} - 4}}{2} \]

6. **Express the final roots**:
The roots can be expressed as:
\[ x_1 = \frac{-1 + i\sqrt{4 - \sqrt{3}}}{2}, \quad x_2 = \frac{-1 - i\sqrt{4 - \sqrt{3}}}{2} \]
where \( i = \sqrt{-1} \).

### Final Answer:
The roots of the equation \( x^2 + x + \frac{1}{\sqrt{3}} = 0 \) are:
\[ x_1 = \frac{-1 + i\sqrt{4 - \sqrt{3}}}{2}, \quad x_2 = \frac{-1 - i\sqrt{4 - \sqrt{3}}}{2} \]