Find the solution set of quadratic equation x² - 8x - 9 = 0 by completing the square method

To solve the quadratic equation \( x^2 - 8x - 9 = 0 \) by completing the square, follow these steps:

### 1. Move the constant term to the right side of the equation.
\[
x^2 - 8x = 9
\]

### 2. Complete the square on the left side.
To complete the square, take half of the coefficient of \( x \), which is \(-8\), divide it by 2 to get \(-4\), and then square it to get \(16\). Add this square to both sides of the equation.

\[
x^2 - 8x + 16 = 9 + 16
\]

\[
(x - 4)^2 = 25
\]

### 3. Solve for \( x \).
Now, take the square root of both sides:

\[
x - 4 = \pm \sqrt{25}
\]

\[
x - 4 = \pm 5
\]

### 4. Find the solution set.
Solve for \( x \) by adding 4 to both sides:

\[
x = 4 \pm 5
\]

This gives us two solutions:

\[
x = 4 + 5 = 9
\]
\[
x = 4 - 5 = -1
\]

### Final Answer:
The solution set is \( x = 9 \) and \( x = -1 \).

Would you like more details on any step, or do you have any questions?

Here are five related questions to deepen your understanding:
1. How would you solve \( x^2 + 6x + 5 = 0 \) by completing the square?
2. Can you explain why we add and subtract the same value when completing the square?
3. What is the general formula for completing the square for any quadratic equation \( ax^2 + bx + c = 0 \)?
4. How does completing the square relate to the quadratic formula?
5. How would you solve \( x^2 - 8x + 16 = 7 \) using the square root method?

**Tip:** When completing the square, always ensure that the coefficient of \( x^2 \) is 1 before proceeding.

Discover how to solve the quadratic equation x² - 8x - 9 = 0 step-by-step using the completing the square method. This problem explores fundamental concepts in quadratic equations and completing the square technique. Suitable for Grades 9-12 students.

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