What is the product all the solutions to the equation `3z^(2)-12z+6=0`?

Options

A. `sqrt(2)`
B. `2`
C. `4`
D. `4sqrt(2)`

Answer

To find the product of all the solutions to the quadratic equation \(3z^2 - 12z + 6 = 0\), we can use the properties of quadratic equations.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given quadratic equation is in the form \(Az^2 + Bz + C = 0\). Here, we identify:
- \(A = 3\)
- \(B = -12\)
- \(C = 6\)

2. **Use the product of roots formula**:
For a quadratic equation \(Az^2 + Bz + C = 0\), the product of the roots (let's denote them as \(\alpha\) and \(\beta\)) can be found using the formula:
\[ \text{Product of roots} = \frac{C}{A} \]

3. **Substitute the values**:
Now, we substitute the values of \(C\) and \(A\) into the formula:
\[ \text{Product of roots} = \frac{C}{A} = \frac{6}{3} \]

4. **Calculate the product**:
Simplifying the fraction gives:
\[ \text{Product of roots} = 2 \]

5. **Conclusion**:
Therefore, the product of all the solutions to the equation \(3z^2 - 12z + 6 = 0\) is \(2\).

### Final Answer:
The product of all the solutions to the equation is \(2\).

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