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Let's go through each part of the problems provided:

### Problem I: Completing the Square

For each quadratic equation, we'll complete the square to solve for \( x \):

1. **(a) \( x^2 + 6x + 3 = 0 \):**
   1. Move the constant term: \( x^2 + 6x = -3 \).
   2. Complete the square: \( \left(\frac{6}{2}\right)^2 = 9 \).
   3. Add 9 to both sides: \( x^2 + 6x + 9 = 6 \).
   4. Rewrite as a square: \( (x + 3)^2 = 6 \).
   5. Solve for \( x \): \( x = -3 \pm \sqrt{6} \).

2. **(b) \( s^2 + 2s - 6 = 0 \):**
   1. Move the constant term: \( s^2 + 2s = 6 \).
   2. Complete the square: \( \left(\frac{2}{2}\right)^2 = 1 \).
   3. Add 1 to both sides: \( s^2 + 2s + 1 = 7 \).
   4. Rewrite as a square: \( (s + 1)^2 = 7 \).
   5. Solve for \( s \): \( s = -1 \pm \sqrt{7} \).

3. **(c) \( x^2 + 4x - 2 = 0 \):**
   1. Move the constant term: \( x^2 + 4x = 2 \).
   2. Complete the square: \( \left(\frac{4}{2}\right)^2 = 4 \).
   3. Add 4 to both sides: \( x^2 + 4x + 4 = 6 \).
   4. Rewrite as a square: \( (x + 2)^2 = 6 \).
   5. Solve for \( x \): \( x = -2 \pm \sqrt{6} \).

4. **(d) \( t^2 - 8t - 5 = 0 \):**
   1. Move the constant term: \( t^2 - 8t = 5 \).
   2. Complete the square: \( \left(\frac{-8}{2}\right)^2 = 16 \).
   3. Add 16 to both sides: \( t^2 - 8t + 16 = 21 \).
   4. Rewrite as a square: \( (t - 4)^2 = 21 \).
   5. Solve for \( t \): \( t = 4 \pm \sqrt{21} \).

5. **(e) \( 2(z + 9) = 1 \):**
   1. Distribute and rearrange: \( 2z + 18 = 1 \Rightarrow z^2 + 

Solve the problems involving quadratic equations, completing the square, and analyzing functions, as shown in the image.

Learn how to solve quadratic equations by completing the square, identify perfect square trinomials, and analyze functions related to real-world applications. Suitable for Grades 8-10.

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