To solve the inequality \(x^2 - 2x - 2 > 0\), we need to follow these steps:

### Step 1: Solve the related equation
First, find the solutions to the equation:

\[
x^2 - 2x - 2 = 0
\]

We can solve this quadratic equation using the quadratic formula:

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

For the equation \(x^2 - 2x - 2 = 0\), the coefficients are:
- \(a = 1\)
- \(b = -2\)
- \(c = -2\)

Substitute these values into the quadratic formula:

\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-2)}}{2(1)}
\]

\[
x = \frac{2 \pm \sqrt{4 + 8}}{2}
\]

\[
x = \frac{2 \pm \sqrt{12}}{2}
\]

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

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

So the solutions are:

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

### Step 2: Analyze the inequality
The solutions to the equation divide the number line into three intervals:
1. \(x < 1 - \sqrt{3}\)
2. \(1 - \sqrt{3} < x < 1 + \sqrt{3}\)
3. \(x > 1 + \sqrt{3}\)

Now, test a point from each interval in the inequality \(x^2 - 2x - 2 > 0\):

- For \(x < 1 - \sqrt{3}\) (e.g., \(x = -5\)): Substituting into the inequality results in a positive value.
- For \(1 - \sqrt{3} < x < 1 + \sqrt{3}\) (e.g., \(x = 0\)): Substituting into the inequality results in a negative value.
- For \(x > 1 + \sqrt{3}\) (e.g., \(x = 5\)): Substituting into the inequality results in a positive value.

### Step 3: Conclusion
The inequality \(x^2 - 2x - 2 > 0\) is satisfied when:

\[
x < 1 - \sqrt{3} \quad \text{or} \quad x > 1 + \sqrt{3}
\]

Thus, the solution is:

\[
x \in (-\infty, 1 - \sqrt{3}) \cup (1 + \sqrt{3}, \infty)
\]

Would you like further details or have any questions about the solution?

Here are 5 related questions to expand your understanding:
1. How would the solution change if the inequality was \(x^2 - 2x - 2 \leq 0\)?
2. Can you graph the function \(f(x) = x^2 - 2x - 2\) and visually verify the solution?
3. What are the real-world applications of solving quadratic inequalities?
4. How does the discriminant \(b^2 - 4ac\) help in determining the number of solutions to a quadratic equation?
5. How can you check the correctness of solutions to inequalities?

**Tip:** When solving quadratic inequalities, always solve the related equation first to find the critical points. Then, test the intervals to determine where the inequality holds.

Learn how to solve the quadratic inequality x^2 - 2x - 2 > 0 using the quadratic formula and interval testing. This detailed step-by-step solution explains the process of solving inequalities with quadratic equations, ideal for students in Grades 9-10.

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