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.