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Let's solve the quadratic equation step by step:

The equation given is:

\[
8(c - 4)^2 - 1 = -1
\]

### Step 1: Simplify the equation
First, add \(1\) to both sides to eliminate the \(-1\) on the left-hand side.

\[
8(c - 4)^2 - 1 + 1 = -1 + 1
\]

This simplifies to:

\[
8(c - 4)^2 = 0
\]

### Step 2: Divide by 8
Now, divide both sides by 8:

\[
(c - 4)^2 = 0
\]

### Step 3: Solve for \(c\)
Since the square of a number equals zero only when the number itself is zero, we get:

\[
c - 4 = 0
\]

### Step 4: Solve for \(c\)
Now, solve for \(c\) by adding 4 to both sides:

\[
c = 4
\]

### Final Answer:
The solution is \(c = 4\).

Would you like more details or have any questions? 

Here are some related questions:
1. What is a quadratic equation, and how can it be identified?
2. How does completing the square method work for solving quadratic equations?
3. Can you explain why squaring both sides preserves equality?
4. What happens when the discriminant of a quadratic equation is zero?
5. Are there other methods to solve quadratic equations apart from factoring and completing the square?

**Tip:** When solving quadratic equations, always check for possible simplifications before applying advanced methods.

Solve the quadratic equation: 8(c - 4)^2 - 1 = -1

This detailed guide demonstrates how to solve the quadratic equation 8(c - 4)^2 - 1 = -1. It covers key algebra concepts such as factoring and the zero-product property, suitable for students in Grades 9-10.

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