If `alpha` and `beta` are the zeroes of the polynomial `x^2-2sqrt3x+3`, then the value of `alpha+beta-alphabeta:`

Options

A. `4sqrt3-3`
B. `2sqrt3-3`
C. `4sqrt3+3`
D. `2sqrt3+3`

Answer

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

### Step-by-Step Solution:

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

2. **Calculate the sum of the zeroes \( \alpha + \beta \)**:
The formula for the sum of the zeroes of a quadratic polynomial \( ax^2 + bx + c \) is given by:
\[ \alpha + \beta = -\frac{b}{a} \]
Substituting the values:
\[ \alpha + \beta = -\frac{-2\sqrt{3}}{1} = 2\sqrt{3} \]

3. **Calculate the product of the zeroes \( \alpha \beta \)**:
The formula for the product of the zeroes is given by:
\[ \alpha \beta = \frac{c}{a} \]
Substituting the values:
\[ \alpha \beta = \frac{3}{1} = 3 \]

4. **Substitute the values into the expression \( \alpha + \beta - \alpha \beta \)**:
Now we need to find:
\[ \alpha + \beta - \alpha \beta = 2\sqrt{3} - 3 \]

5. **Final result**:
Thus, the value of \( \alpha + \beta - \alpha \beta \) is:
\[ 2\sqrt{3} - 3 \]