The given quadratic equation is:

\[
3x^2 + 11x + 8 = 0
\]

We will solve this equation using the **quadratic formula**:

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

In your equation, \(a = 3\), \(b = 11\), and \(c = 8\). Let's plug these values into the formula:

\[
x = \frac{-11 \pm \sqrt{11^2 - 4(3)(8)}}{2(3)}
\]

Now let's calculate the discriminant:

\[
11^2 = 121
\]
\[
4(3)(8) = 96
\]
\[
\text{Discriminant} = 121 - 96 = 25
\]

Since the discriminant is positive, there are two real solutions. Let's continue:

\[
x = \frac{-11 \pm \sqrt{25}}{6}
\]
\[
x = \frac{-11 \pm 5}{6}
\]

Now, solving for the two values of \(x\):

1. \(x_1 = \frac{-11 + 5}{6} = \frac{-6}{6} = -1\)
2. \(x_2 = \frac{-11 - 5}{6} = \frac{-16}{6} = \frac{-8}{3}\)

Thus, the solutions are:

\[
x_1 = -1 \quad \text{and} \quad x_2 = \frac{-8}{3}
\]

Would you like more details or have any questions?

Here are some related questions:
1. How do we determine if a quadratic equation has real solutions?
2. What is the significance of the discriminant in the quadratic formula?
3. Can quadratic equations have complex solutions?
4. What are some other methods for solving quadratic equations?
5. How does factoring compare to using the quadratic formula?

**Tip:** Always check the discriminant before solving a quadratic equation to know the nature of the solutions (real or complex).

Learn how to solve the quadratic equation 3x^2 + 11x + 8 with a detailed step-by-step solution. This problem covers key algebra concepts, including the quadratic formula and discriminant calculation. Suitable for students in Grades 8-10.

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