Solve the equation.
3xsquaredplus 3xplus 2equals0
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Part 1
The solution set is StartSet nothing EndSet
  
.
​(Simplify your answer. Type an exact​ answer, using radicals and i as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as​ needed.)

Let's solve the quadratic equation:

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

We will solve this using the quadratic formula, which is:

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

In this equation, we have:
- \( a = 3 \)
- \( b = 3 \)
- \( c = 2 \)

### Step 1: Calculate the discriminant
The discriminant is \( b^2 - 4ac \):

\[
\Delta = 3^2 - 4(3)(2) = 9 - 24 = -15
\]

Since the discriminant is negative, the solutions will involve imaginary numbers.

### Step 2: Apply the quadratic formula
Now substitute the values of \( a \), \( b \), and \( \Delta \) into the quadratic formula:

\[
x = \frac{-3 \pm \sqrt{-15}}{2(3)} = \frac{-3 \pm \sqrt{15}i}{6}
\]

### Step 3: Simplify the solution
The two solutions are:

\[
x = \frac{-3 + \sqrt{15}i}{6} \quad \text{and} \quad x = \frac{-3 - \sqrt{15}i}{6}
\]

So, the solution set is:

\[
x = \frac{-1}{2} \pm \frac{\sqrt{15}}{6}i
\]

This represents the two complex solutions of the quadratic equation.

---

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. What are the conditions for a quadratic equation to have real versus imaginary solutions?
2. Can you explain how the discriminant affects the nature of the roots?
3. What is the significance of the quadratic formula in solving all forms of quadratic equations?
4. How do we interpret complex solutions geometrically?
5. What other methods can be used to solve quadratic equations, besides the quadratic formula?

**Tip:** When the discriminant is negative, the quadratic equation has no real solutions, only complex ones.

Solve the equation. 3x^2 + 3x + 2 = 0. (Simplify your answer. Type an exact​ answer, using radicals and i as needed. Use integers or fractions for any numbers in the expression. Use a comma to separate answers as​ needed.)

Learn how to solve the quadratic equation 3x^2 + 3x + 2 = 0 using the quadratic formula. This problem involves calculating the discriminant, identifying complex solutions, and simplifying the result. Suitable for students in Grades 9-12.

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