Mathematics worksheet focusing on factoring perfect square trinomials, including fill-in-the-blank and matching exercises.

Let's solve the worksheet step by step.

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A. Complete the following expression to make a perfect square trinomial

A perfect square trinomial has the form:

$$
a^2 + 2ab + b^2 = (a + b)^2
$$

or
$$
a^2 - 2ab + b^2 = (a - b)^2
$$

To complete the square, we use the formula:
> Take half of the coefficient of the middle term, then square it.

So for $ a^2 + bx + \_\_\_\_ $, the missing constant is:
$$
\left(\frac{b}{2}\right)^2
$$

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#### Solving each:

1) $ a^2 + 8a + \_\_\_\_ $
Half of 8 is 4 → $ 4^2 = 16 $
✔ 16

2) $ a^2 - 10a + \_\_\_\_ $
Half of -10 is -5 → $ (-5)^2 = 25 $
✔ 25

3) $ a^2 + 4a + \_\_\_\_ $
Half of 4 is 2 → $ 2^2 = 4 $
✔ 4

4) $ a^2 - 24a + \_\_\_\_ $
Half of -24 is -12 → $ (-12)^2 = 144 $
✔ 144

5) $ a^2 - 4a + \_\_\_\_ $
Half of -4 is -2 → $ (-2)^2 = 4 $
✔ 4

6) $ a^2 + 24a + \_\_\_\_ $
Half of 24 is 12 → $ 12^2 = 144 $
✔ 144

7) $ a^2 + 12a + \_\_\_\_ $
Half of 12 is 6 → $ 6^2 = 36 $
✔ 36

8) $ a^2 - 12a + \_\_\_\_ $
Half of -12 is -6 → $ (-6)^2 = 36 $
✔ 36

9) $ a^2 - 6a + \_\_\_\_ $
Half of -6 is -3 → $ (-3)^2 = 9 $
✔ 9

10) $ a^2 + 10a + \_\_\_\_ $
Half of 10 is 5 → $ 5^2 = 25 $
✔ 25

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✔ Answers for Part A:

1) 16
2) 25
3) 4
4) 144
5) 4
6) 144
7) 36
8) 36
9) 9
10) 25

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B. Match Column A with its factor in Column B

We will factor each expression in Column A and match it with the correct squared binomial in Column B.

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1) $ a^2 + 8a + 16 $
→ $ (a + 4)^2 $ because $ 4^2 = 16 $, and $ 2 \cdot a \cdot 4 = 8a $
✔ Matches: (a + 4)²

2) $ a^2 - 10a + 25 $
→ $ (a - 5)^2 $ because $ (-5)^2 = 25 $, $ 2 \cdot a \cdot (-5) = -10a $
✔ Matches: (a - 5)²

3) $ 4a^2 - 4a + 1 $
This is $ (2a)^2 - 2 \cdot 2a \cdot 1 + 1^2 = (2a - 1)^2 $
✔ Matches: (2a - 1)²

4) $ 4a^2 - 12a + 9 $
$ (2a)^2 - 2 \cdot 2a \cdot 3 + 3^2 = (2a - 3)^2 $
✔ Matches: (2a - 3)²

5) $ 4a^2 + 4a + 1 $
$ (2a)^2 + 2 \cdot 2a \cdot 1 + 1^2 = (2a + 1)^2 $
✔ Matches: (2a + 1)²

6) $ 9a^2 + 24a + 16 $
$ (3a)^2 + 2 \cdot 3a \cdot 4 + 4^2 = (3a + 4)^2 $
✔ Matches: (3a + 4)²

7) $ a^2 + 12a + 36 $
$ (a + 6)^2 $ because $ 6^2 = 36 $, $ 2 \cdot a \cdot 6 = 12a $
✔ Matches: (a + 6)²

8) $ a^2 - 12a + 36 $
$ (a - 6)^2 $ because $ (-6)^2 = 36 $, $ 2 \cdot a \cdot (-6) = -12a $
✔ Matches: (a - 6)²

9) $ a^2 - 8a + 16 $
$ (a - 4)^2 $ because $ (-4)^2 = 16 $, $ 2 \cdot a \cdot (-4) = -8a $
✔ Matches: (a - 4)²

10) $ a^2 + 10a + 25 $
$ (a + 5)^2 $ because $ 5^2 = 25 $, $ 2 \cdot a \cdot 5 = 10a $
✔ Matches: (a + 5)²

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✔ Matching for Part B:

| Column A | Matched to Column B |
|---------|---------------------|
| 1. $ a^2 + 8a + 16 $ | $ (a + 4)^2 $ |
| 2. $ a^2 - 10a + 25 $ | $ (a - 5)^2 $ |
| 3. $ 4a^2 - 4a + 1 $ | $ (2a - 1)^2 $ |
| 4. $ 4a^2 - 12a + 9 $ | $ (2a - 3)^2 $ |
| 5. $ 4a^2 + 4a + 1 $ | $ (2a + 1)^2 $ |
| 6. $ 9a^2 + 24a + 16 $ | $ (3a + 4)^2 $ |
| 7. $ a^2 + 12a + 36 $ | $ (a + 6)^2 $ |
| 8. $ a^2 - 12a + 36 $ | $ (a - 6)^2 $ |
| 9. $ a^2 - 8a + 16 $ | $ (a - 4)^2 $ |
| 10. $ a^2 + 10a + 25 $ | $ (a + 5)^2 $ |

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✔ Final Answers:

#### Part A:
1) 16
2) 25
3) 4
4) 144
5) 4
6) 144
7) 36
8) 36
9) 9
10) 25

#### Part B: Matching
- 1 → (a + 4)²
- 2 → (a - 5)²
- 3 → (2a - 1)²
- 4 → (2a - 3)²
- 5 → (2a + 1)²
- 6 → (3a + 4)²
- 7 → (a + 6)²
- 8 → (a - 6)²
- 9 → (a - 4)²
- 10 → (a + 5)²

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