The sum and the product of the roots of the quadratic equations ` 2 x^(2) + 14 x + 24 = 0 `

Answer

To find the sum and the product of the roots of the quadratic equation \(2x^2 + 14x + 24 = 0\), we can use the relationships derived from Vieta's formulas.

### Step-by-Step Solution:

1. **Identify coefficients**:
The given quadratic equation is in the standard form \(ax^2 + bx + c = 0\). Here:
- \(a = 2\)
- \(b = 14\)
- \(c = 24\)

2. **Calculate the sum of the roots**:
The sum of the roots (\(\alpha + \beta\)) of a quadratic equation can be calculated using the formula:
\[ \alpha + \beta = -\frac{b}{a} \]
Substituting the values of \(b\) and \(a\):
\[ \alpha + \beta = -\frac{14}{2} = -7 \]

3. **Calculate the product of the roots**:
The product of the roots (\(\alpha \beta\)) can be calculated using the formula:
\[ \alpha \beta = \frac{c}{a} \]
Substituting the values of \(c\) and \(a\):
\[ \alpha \beta = \frac{24}{2} = 12 \]

4. **Final Result**:
Therefore, the sum of the roots is \(-7\) and the product of the roots is \(12\).

### Summary:
- Sum of the roots (\(\alpha + \beta\)) = \(-7\)
- Product of the roots (\(\alpha \beta\)) = \(12\)