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The exercise focuses on completing the square to rewrite quadratic trinomials in the form \((x + b)^2 + c\). Let’s go through the first example provided, and then we can apply the steps to the exercises in parts (a) through (f).

### Example Analysis
The given equation is:
\[
x^2 - 6x - 8
\]

To complete the square:
1. Take half of the coefficient of \(x\) (which is \(-6\)), divide by 2 to get \(-3\), then square it to get \(9\).
2. Add and subtract \(9\) within the expression:
   \[
   x^2 - 6x - 8 = (x^2 - 6x + 9) - 9 - 8
   \]
3. This simplifies to:
   \[
   (x - 3)^2 - 17
   \]

Now, let's apply this method to the exercises:

### Solution for Each Part

#### (a) \( x^2 - 2x - 8 \)
1. Take half of \(-2\), square it to get \(1\).
2. Rewrite as:
   \[
   x^2 - 2x - 8 = (x^2 - 2x + 1) - 1 - 8 = (x - 1)^2 - 9
   \]

#### (b) \( x^2 + 4x - 1 \)
1. Take half of \(4\), square it to get \(4\).
2. Rewrite as:
   \[
   x^2 + 4x - 1 = (x^2 + 4x + 4) - 4 - 1 = (x + 2)^2 - 5
   \]

#### (c) \( x^2 + 10x + 3 \)
1. Take half of \(10\), square it to get \(25\).
2. Rewrite as:
   \[
   x^2 + 10x + 3 = (x^2 + 10x + 25) - 25 + 3 = (x + 5)^2 - 22
   \]

#### (d) \( x^2 - 16x - 3 \)
1. Take half of \(-16\), square it to get \(64\).
2. Rewrite as:
   \[
   x^2 - 16x - 3 = (x^2 - 16x + 64) - 64 - 3 = (x - 8)^2 - 67
   \]

#### (e) \( x^2 + 18x + 7 \)
1. Take half of \(18\), square it to get \(81\).
2. Rewrite as:
   \[
   x^2 + 18x + 7 = (x^2 + 18x + 81) - 81 + 7 = (x + 9)^2 - 74
   \]

#### (f) \( x^2 - 32x - 1 \)
1. Take half of \(-32\), square it to get \(256\).
2. Rewrite as:
   \[
   x^2 - 32x - 1 = (x^2 - 32x + 256) - 256 - 1 = (x - 16)^2 - 257
   \]

### Summary
Here are the completed square forms for each part:
- (a) \( (x - 1)^2 - 9 \)
- (b) \( (x + 2)^2 - 5 \)
- (c) \( (x + 5)^2 - 22 \)
- (d) \( (x - 8)^2 - 67 \)
- (e) \( (x + 9)^2 - 74 \)
- (f) \( (x - 16)^2 - 257 \)

Would you like further explanations on any of these steps?

### Follow-up Questions
1. How can you use the method of completing the square to solve quadratic equations?
2. Why is completing the square useful for converting a quadratic expression into vertex form?
3. How does completing the square relate to the formula for the vertex of a parabola?
4. Can completing the square help in integrating quadratic expressions?
5. How would the process differ if the coefficient of \(x^2\) were not 1?

### Tip
When completing the square, remember that you only add and subtract the squared term to form a perfect square, keeping the expression balanced.

Complete the square for the given quadratic expressions: (a) x^2 - 2x - 8, (b) x^2 + 4x - 1, (c) x^2 + 10x + 3, (d) x^2 - 16x - 3, (e) x^2 + 18x + 7, (f) x^2 - 32x - 1.

Learn how to complete the square for quadratic expressions such as x^2 - 2x - 8 and x^2 + 10x + 3 with detailed step-by-step explanations. This method is useful for rewriting quadratic trinomials in the form (x + b)^2 + c and is essential for students in Grade 9.

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