To solve the quadratic equation \(9x^2 - 6x - 11 = 0\), we can use the quadratic formula:

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

where \(a = 9\), \(b = -6\), and \(c = -11\). Substituting these values into the formula:

\[
x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(9)(-11)}}{2(9)}
\]

Simplify inside the square root:

\[
x = \frac{6 \pm \sqrt{36 + 396}}{18}
\]

\[
x = \frac{6 \pm \sqrt{432}}{18}
\]

Next, simplify the square root:

\[
\sqrt{432} = \sqrt{144 \times 3} = 12\sqrt{3}
\]

So, the equation becomes:

\[
x = \frac{6 \pm 12\sqrt{3}}{18}
\]

Simplify the fraction:

\[
x = \frac{6}{18} \pm \frac{12\sqrt{3}}{18}
\]

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

Thus, the solutions are:

\[
x_1 = \frac{1}{3} + \frac{2\sqrt{3}}{3}, \quad x_2 = \frac{1}{3} - \frac{2\sqrt{3}}{3}
\]

Would you like a step-by-step breakdown or have any questions?

Here are some related questions you might consider:
1. How do you simplify square roots in quadratic equations?
2. What are other methods to solve quadratic equations besides the quadratic formula?
3. How can you verify the solutions of a quadratic equation?
4. What does it mean graphically when solving \(9x^2 - 6x - 11 = 0\)?
5. How does the discriminant \(b^2 - 4ac\) affect the nature of the solutions?

**Tip:** Always simplify radicals to their simplest form to make solving easier!

Explore how to solve the quadratic equation 9x^2 - 6x - 11 with a step-by-step breakdown using the quadratic formula. This problem introduces core algebraic concepts like simplifying radicals and analyzing the discriminant. Perfect for high school students in Grades 9-11.

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