If one root of the quadratic equation `x^(2) + 6x + 2 = 0` is, `(2)/(3)` then find the other root of the equation.

Answer

To find the other root of the quadratic equation \(x^2 + 6x + 2 = 0\) given that one root is \(\frac{2}{3}\), we can use the relationships between the roots of a quadratic equation.

### Step-by-Step Solution:

1. **Identify the coefficients**:
For the quadratic equation \(x^2 + 6x + 2 = 0\), we have:
- \(a = 1\)
- \(b = 6\)
- \(c = 2\)

2. **Use the sum of roots formula**:
The sum of the roots \(\alpha + \beta\) of a quadratic equation is given by the formula:
\[ \alpha + \beta = -\frac{b}{a} \]
Substituting the values:
\[ \alpha + \beta = -\frac{6}{1} = -6 \]
Since one root \(\alpha = \frac{2}{3}\), we can find the other root \(\beta\):
\[ \frac{2}{3} + \beta = -6 \]

3. **Solve for \(\beta\)**:
Rearranging the equation to find \(\beta\):
\[ \beta = -6 - \frac{2}{3} \]
To combine these, convert \(-6\) into a fraction:
\[ -6 = -\frac{18}{3} \]
Thus,
\[ \beta = -\frac{18}{3} - \frac{2}{3} = -\frac{20}{3} \]

4. **Conclusion**:
The other root of the equation is:
\[ \beta = -\frac{20}{3} \]

### Final Answer:
The other root of the equation is \(-\frac{20}{3}\).