The expression you've provided is a quadratic equation of the form \( ax^2 + bx + c \), where \( a = 3 \), \( b = 11 \), and \( c = 6 \).

To solve this quadratic equation for \( x \), we can use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

### Step-by-Step Solution:

1. **Identify the coefficients:**
   \[
   a = 3, \quad b = 11, \quad c = 6
   \]

2. **Calculate the discriminant:**
   \[
   \text{Discriminant} = b^2 - 4ac = 11^2 - 4(3)(6) = 121 - 72 = 49
   \]

3. **Take the square root of the discriminant:**
   \[
   \sqrt{49} = 7
   \]

4. **Apply the quadratic formula:**
   \[
   x = \frac{-11 \pm 7}{2(3)}
   \]

5. **Find the two possible solutions:**

   - For the plus sign (\(+\)):
     \[
     x_1 = \frac{-11 + 7}{6} = \frac{-4}{6} = -\frac{2}{3}
     \]

   - For the minus sign (\(-\)):
     \[
     x_2 = \frac{-11 - 7}{6} = \frac{-18}{6} = -3
     \]

### Solutions:

\[
x = -\frac{2}{3} \quad \text{or} \quad x = -3
\]

Would you like more details or have any questions?

Here are five related questions:

1. How would you factor the quadratic equation \( 3x^2 + 11x + 6 \)?
2. Can you explain the significance of the discriminant in determining the nature of the roots?
3. How does the quadratic formula change if the discriminant is negative?
4. What are the roots of the equation \( 3x^2 - 11x + 6 \)?
5. How would you graph the equation \( 3x^2 + 11x + 6 = 0 \) on the coordinate plane?

**Tip:** The discriminant helps you determine whether the quadratic equation has real or complex roots.

Learn how to solve the quadratic equation 3x² + 11x + 6 with a detailed step-by-step solution using the quadratic formula. This problem covers key algebraic concepts and the use of discriminant to determine the nature of roots. Suitable for high school students.

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