To solve the quadratic equation by completing the square, let's go step by step.

Given equation:

$x^2 - 2x - 9 = 0$

Step 1: Move the constant term to the other side.

$x^2 - 2x = 9$

Step 2: Complete the square.

To complete the square, take half of the coefficient of $x$, square it, and add it to both sides. The coefficient of $x$ is $-2$, so: $\left(\frac{-2}{2}\right)^2 = 1$ Now, add 1 to both sides: $x^2 - 2x + 1 = 9 + 1$ $(x - 1)^2 = 10$

Step 3: Solve for $x$.

Take the square root of both sides: $x - 1 = \pm \sqrt{10}$ Now, solve for $x$: $x = 1 \pm \sqrt{10}$

Final Answer:

The solutions are: $x = 1 + \sqrt{10}, \quad x = 1 - \sqrt{10}$

Do you need more details or have any questions?

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Here are 5 related questions for further practice:

1. Solve $x^2 - 4x - 5 = 0$ by completing the square.
2. Solve $x^2 + 6x - 7 = 0$ by completing the square.
3. What is the relationship between completing the square and the quadratic formula?
4. How can completing the square help in graphing quadratic functions?
5. Can all quadratic equations be solved by completing the square? Why or why not?

Tip: Completing the square is a method that can also be used to convert quadratic equations into vertex form, making it easier to analyze the vertex of the parabola.