If `alphaandbeta` are the zeroes of a polynomial `x^(2)-4sqrt(3)x+3`,then find the value of `alpha+beta-alphabeta`.

Options

A. `4sqrt(3)-3`
B. `sqrt(3)-3`
C. `-3-4sqrt(3)`
D. `4sqrt(3)`

Answer

To solve the problem, we need to find the value of \( \alpha + \beta - \alpha \beta \), where \( \alpha \) and \( \beta \) are the zeros of the polynomial \( x^2 - 4\sqrt{3}x + 3 \).

### Step-by-step Solution:

1. **Identify the coefficients of the polynomial**:
The polynomial is given as \( x^2 - 4\sqrt{3}x + 3 \). Here, we can identify:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -4\sqrt{3} \) (coefficient of \( x \))
- \( c = 3 \) (constant term)

2. **Calculate \( \alpha + \beta \)**:
The sum of the roots \( \alpha + \beta \) can be calculated using the formula:
\[ \alpha + \beta = -\frac{b}{a} \]
Substituting the values of \( b \) and \( a \):
\[ \alpha + \beta = -\frac{-4\sqrt{3}}{1} = 4\sqrt{3} \]

3. **Calculate \( \alpha \beta \)**:
The product of the roots \( \alpha \beta \) can be calculated using the formula:
\[ \alpha \beta = \frac{c}{a} \]
Substituting the values of \( c \) and \( a \):
\[ \alpha \beta = \frac{3}{1} = 3 \]

4. **Calculate \( \alpha + \beta - \alpha \beta \)**:
Now, we need to find \( \alpha + \beta - \alpha \beta \):
\[ \alpha + \beta - \alpha \beta = 4\sqrt{3} - 3 \]

5. **Final Answer**:
Therefore, the value of \( \alpha + \beta - \alpha \beta \) is:
\[ \boxed{4\sqrt{3} - 3} \]