To solve the quadratic equation \( 3x^2 - 9x - 2 = 0 \), we can use the quadratic formula:

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

For the equation \( 3x^2 - 9x - 2 = 0 \), the coefficients are:
- \( a = 3 \)
- \( b = -9 \)
- \( c = -2 \)

Now, plug these values into the quadratic formula:

\[
x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(3)(-2)}}{2(3)}
\]

Simplify the expression:

\[
x = \frac{9 \pm \sqrt{81 + 24}}{6}
\]
\[
x = \frac{9 \pm \sqrt{105}}{6}
\]

This gives us two solutions:

\[
x_1 = \frac{9 + \sqrt{105}}{6}
\]
\[
x_2 = \frac{9 - \sqrt{105}}{6}
\]

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

\[
x_1 \approx 2.79 \quad \text{and} \quad x_2 \approx -0.12
\]

Would you like further details or have any questions?

Here are some related questions:
1. How do you derive the quadratic formula?
2. What does the discriminant tell us about the nature of the roots?
3. How can you factor a quadratic equation?
4. What is the significance of the coefficients \(a\), \(b\), and \(c\) in a quadratic equation?
5. Can all quadratic equations be solved using the quadratic formula?

**Tip:** Always check the discriminant \(b^2 - 4ac\) before applying the quadratic formula, as it indicates whether the roots are real, complex, or repeated.

Learn how to solve the quadratic equation 3x^2 - 9x - 2 = 0 step-by-step. This problem covers key algebraic concepts, including the quadratic formula and discriminant. Suitable for students in Grades 10-12.

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