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

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

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

Substitute these values into the quadratic formula:

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

Simplifying further:

\[
x = \frac{2 \pm \sqrt{4 + 8}}{2}
\]
\[
x = \frac{2 \pm \sqrt{12}}{2}
\]
\[
x = \frac{2 \pm 2\sqrt{3}}{2}
\]

Now, simplify the expression:

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

So the exact solutions are:

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

For the decimal approximations:

\[
x \approx 2.73 \quad \text{or} \quad x \approx -0.73
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How would the solutions change if the coefficient of \(x\) was positive?
2. What is the significance of the discriminant in this equation?
3. How can you verify the solutions by substitution?
4. What are some alternative methods to solve this quadratic equation?
5. How does changing the constant term affect the roots?

**Tip:** When simplifying square roots, look for perfect squares inside the radical to simplify the expression.

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

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