Let's solve the quadratic equation \(3x^2 - 2x - 2 = 0\) using the quadratic formula:

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

For the equation \(3x^2 - 2x - 2 = 0\), the coefficients are:
- \(a = 3\)
- \(b = -2\)
- \(c = -2\)

### Step 1: Calculate the discriminant
The discriminant \(\Delta\) is given by:

\[
\Delta = b^2 - 4ac = (-2)^2 - 4(3)(-2) = 4 + 24 = 28
\]

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

\[
x = \frac{-(-2) \pm \sqrt{28}}{2(3)} = \frac{2 \pm \sqrt{28}}{6}
\]

Simplify \(\sqrt{28}\):
\[
\sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7}
\]

Thus, the solutions are:

\[
x = \frac{2 \pm 2\sqrt{7}}{6} = \frac{1 \pm \sqrt{7}}{3}
\]

### Final Answer:
The two solutions for \(x\) are:

\[
x_1 = \frac{1 + \sqrt{7}}{3} \quad \text{and} \quad x_2 = \frac{1 - \sqrt{7}}{3}
\]

These are the exact solutions. If you need approximate decimal values:

\[
x_1 \approx 1.215 \quad \text{and} \quad x_2 \approx -0.548
\]

Would you like further details or have any questions?

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Here are five related questions to consider:
1. How can you verify the solutions by substituting them back into the original equation?
2. What is the significance of the discriminant in this solution?
3. How would you graph the function \(y = 3x^2 - 2x - 2\) and find its roots?
4. Can this quadratic equation be solved using a method like completing the square?
5. How does the coefficient of \(x^2\) affect the shape of the parabola?

**Tip:** Simplifying square roots when possible, like \(\sqrt{28}\) to \(2\sqrt{7}\), makes your final answer clearer and more concise.

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

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