Sure! Let's solve the quadratic equation \( x^2 - 6x - 11 = 0 \) by completing the square.

### Step 1: Move the constant term to the other side
Start by moving the constant term \(-11\) to the right side of the equation:

\[
x^2 - 6x = 11
\]

### Step 2: Complete the square
To complete the square, we need to add and subtract a specific value on the left side. This value is found by taking half of the coefficient of \(x\) (which is \(-6\)), squaring it, and adding it to both sides.

\[
\text{Half of } -6 = -3
\]
\[
(-3)^2 = 9
\]

So, add 9 to both sides of the equation:

\[
x^2 - 6x + 9 = 11 + 9
\]

### Step 3: Simplify both sides
Now, the left side is a perfect square trinomial:

\[
(x - 3)^2 = 20
\]

### Step 4: Take the square root of both sides
Next, take the square root of both sides to solve for \(x\):

\[
x - 3 = \pm \sqrt{20}
\]

Since \(\sqrt{20}\) can be simplified to \(2\sqrt{5}\), we have:

\[
x - 3 = \pm 2\sqrt{5}
\]

### Step 5: Solve for \(x\)
Finally, solve for \(x\) by adding 3 to both sides:

\[
x = 3 \pm 2\sqrt{5}
\]

So the solutions are:

\[
x_1 = 3 + 2\sqrt{5}, \quad x_2 = 3 - 2\sqrt{5}
\]

These are the same solutions we found using the quadratic formula.

Would you like more details or have any questions?

Here are some related questions you might explore:

1. Why is completing the square useful in solving quadratic equations?
2. How does completing the square relate to the quadratic formula?
3. What is the vertex form of a quadratic function, and how is it derived by completing the square?
4. Can completing the square be used to solve any quadratic equation?
5. How is completing the square used in the derivation of the quadratic formula?
6. How can you complete the square when the coefficient of \(x^2\) is not 1?
7. What are some practical applications of completing the square in real-world problems?
8. How does completing the square help in understanding the geometry of a parabola?

**Tip:** Completing the square is not only a method for solving quadratics but also a key technique in integration and in understanding the geometry of conic sections.

Learn how to solve the quadratic equation x^2 - 6x - 11 = 0 by completing the square. Follow a detailed step-by-step solution including the application of the quadratic formula and completing the square theorem. Suitable for high school students studying algebra.

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