Let's work through the quadratic equation \(3x^2 + 11x + 6 = 0\) again, step by step, to solve for \(x\).

### 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:**
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-11 \pm 7}{6}
   \]

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

### Final Solutions:

\[
x_1 = -\frac{2}{3} \quad \text{and} \quad x_2 = -3
\]

These are the roots of the equation \(3x^2 + 11x + 6 = 0\).

Would you like further details or have any questions?

Here are five related questions:

1. What is the vertex form of the quadratic equation \(3x^2 + 11x + 6\)?
2. How would you factor the quadratic equation \(3x^2 + 11x + 6\) directly?
3. What is the axis of symmetry for the parabola described by the equation \(3x^2 + 11x + 6 = 0\)?
4. How would you determine the y-intercept of the quadratic function \(3x^2 + 11x + 6\)?
5. What are the roots of the equation \(3x^2 + 11x - 6 = 0\)?

**Tip:** When solving quadratic equations, checking if they can be factored easily can save time compared to using the quadratic formula.

Learn how to solve the quadratic equation 3x^2 + 11x + 6 = 0 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the discriminant and quadratic formula. Suitable for high school students in Grades 10-12.

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