Solve the quadratic equation by completing the square. (Enter your answers as a comma-separated list.)
x2 − 12x − 64 = 0

To solve the quadratic equation \( x^2 - 12x - 64 = 0 \) by completing the square, follow these steps:

### Step 1: Move the constant term to the other side of the equation
We start with:
\[
x^2 - 12x - 64 = 0
\]
Add 64 to both sides:
\[
x^2 - 12x = 64
\]

### Step 2: Complete the square
To complete the square, take half of the coefficient of \( x \), square it, and add it to both sides of the equation. The coefficient of \( x \) is -12, so:
\[
\left(\frac{-12}{2}\right)^2 = (-6)^2 = 36
\]
Now add 36 to both sides:
\[
x^2 - 12x + 36 = 64 + 36
\]
\[
x^2 - 12x + 36 = 100
\]

### Step 3: Write the left-hand side as a perfect square
The left-hand side is now a perfect square trinomial:
\[
(x - 6)^2 = 100
\]

### Step 4: Solve for \( x \)
Take the square root of both sides:
\[
x - 6 = \pm \sqrt{100}
\]
\[
x - 6 = \pm 10
\]
Now solve for \( x \) by adding 6 to both sides:
\[
x = 6 + 10 \quad \text{or} \quad x = 6 - 10
\]
\[
x = 16 \quad \text{or} \quad x = -4
\]

### Final Answer:
The solutions are:
\[
x = 16, -4
\]

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

Here are 5 questions for further practice:
1. Solve \( x^2 + 8x + 7 = 0 \) by completing the square.
2. Complete the square for \( x^2 - 10x - 25 = 0 \).
3. How do you recognize a perfect square trinomial?
4. What is the general formula for completing the square?
5. How can completing the square help derive the quadratic formula?

**Tip:** When completing the square, always remember to balance the equation by adding the same value to both sides.

Solve the quadratic equation by completing the square. x^2 − 12x − 64 = 0

Learn how to solve the quadratic equation x^2 − 12x − 64 using the completing the square method. Follow the step-by-step process to master this key algebraic technique, suitable for students in Grades 8-10.

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