If the product of zeroes of the polynomial `(ax^(2)-6x-6)` is 4, find the value of a.

Explanation

To solve the problem, we need to find the value of \( a \) in the polynomial \( ax^2 - 6x - 6 \) given that the product of its zeroes is 4.

### Step-by-Step Solution:

1. **Identify the polynomial and its coefficients**:
The polynomial is given as:
\[ P(x) = ax^2 - 6x - 6 \]
Here, we can identify the coefficients:
- \( A = a \)
- \( B = -6 \)
- \( C = -6 \)

2. **Recall the formula for the product of the zeroes**:
For a quadratic polynomial \( Ax^2 + Bx + C \), the product of the zeroes (let's denote them as \( \alpha \) and \( \beta \)) is given by:
\[ \text{Product of zeroes} = \frac{C}{A} \]

3. **Set up the equation using the given product of zeroes**:
We know from the problem that the product of the zeroes is 4. Therefore, we can set up the equation:
\[ \frac{C}{A} = 4 \]
Substituting the values of \( C \) and \( A \):
\[ \frac{-6}{a} = 4 \]

4. **Solve for \( a \)**:
To find \( a \), we can cross-multiply:
\[ -6 = 4a \]
Now, divide both sides by 4:
\[ a = \frac{-6}{4} \]
Simplifying this gives:
\[ a = -\frac{3}{2} \]

5. **Conclusion**:
The value of \( a \) is:
\[ a = -\frac{3}{2} \]