The roots of the equation `sqrt3x^(2) - 2x - sqrt3 = 0` are :

Options

A. `-sqrt3 , (1)/(sqrt3)`
B. `2 , 3`
C. `(sqrt3)/(2) , - (2)/(sqrt3)`
D. `sqrt3 , - (1)/(sqrt3)`

Answer

To find the roots of the equation \( \sqrt{3}x^2 - 2x - \sqrt{3} = 0 \), we can use the method of factoring or the quadratic formula. Here, we will use factoring.

### Step-by-Step Solution:

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

2. **Multiply \( a \) and \( c \)**:
Calculate \( ac \):
\[ ac = \sqrt{3} \cdot (-\sqrt{3}) = -3 \]

3. **Find two numbers that multiply to \( ac \) and add to \( b \)**:
We need two numbers that multiply to \(-3\) (the value of \( ac \)) and add up to \(-2\) (the value of \( b \)). The numbers are:
\[ 1 \quad \text{and} \quad -3 \]
Because:
\[ 1 \cdot (-3) = -3 \quad \text{and} \quad 1 + (-3) = -2 \]

4. **Rewrite the middle term**:
Rewrite the equation by splitting the middle term using the numbers found:
\[ \sqrt{3}x^2 + x - 3x - \sqrt{3} = 0 \]

5. **Group the terms**:
Group the terms:
\[ (\sqrt{3}x^2 + x) + (-3x - \sqrt{3}) = 0 \]

6. **Factor by grouping**:
Factor out common terms from each group:
\[ x(\sqrt{3}x + 1) - \sqrt{3}(3x + 1) = 0 \]

7. **Combine the factors**:
Combine the factors:
\[ (\sqrt{3}x + 1)(x - \sqrt{3}) = 0 \]

8. **Set each factor to zero**:
Set each factor to zero to find the roots:
\[ \sqrt{3}x + 1 = 0 \quad \text{or} \quad x - \sqrt{3} = 0 \]

9. **Solve for \( x \)**:
For the first equation:
\[ \sqrt{3}x + 1 = 0 \implies \sqrt{3}x = -1 \implies x = -\frac{1}{\sqrt{3}} \]
For the second equation:
\[ x - \sqrt{3} = 0 \implies x = \sqrt{3} \]

10. **Final roots**:
The roots of the equation are:
\[ x = \sqrt{3} \quad \text{and} \quad x = -\frac{1}{\sqrt{3}} \]

### Summary of Roots:
The roots of the equation \( \sqrt{3}x^2 - 2x - \sqrt{3} = 0 \) are:
- \( x = \sqrt{3} \)
- \( x = -\frac{1}{\sqrt{3}} \)