Worksheet showing examples and solutions for factoring the difference of squares in algebra.

To solve the problems involving factoring the difference of squares, we need to use the formula:

\[
a^2 - b^2 = (a - b)(a + b)
\]

This formula states that the difference of two squares can be factored into the product of the sum and the difference of their square roots.

Let's go through each problem step by step.

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Section 1: Factoring Difference of Squares

#### Problem 1: \( x^2 - 4 \)
- Identify \( a \) and \( b \):
\[
a^2 = x^2 \quad \text{and} \quad b^2 = 4
\]
So, \( a = x \) and \( b = 2 \).
- Apply the formula:
\[
x^2 - 4 = (x - 2)(x + 2)
\]

#### Problem 2: \( x^2 - 9 \)
- Identify \( a \) and \( b \):
\[
a^2 = x^2 \quad \text{and} \quad b^2 = 9
\]
So, \( a = x \) and \( b = 3 \).
- Apply the formula:
\[
x^2 - 9 = (x - 3)(x + 3)
\]

#### Problem 3: \( x^2 - 16 \)
- Identify \( a \) and \( b \):
\[
a^2 = x^2 \quad \text{and} \quad b^2 = 16
\]
So, \( a = x \) and \( b = 4 \).
- Apply the formula:
\[
x^2 - 16 = (x - 4)(x + 4)
\]

#### Problem 4: \( x^2 - 25 \)
- Identify \( a \) and \( b \):
\[
a^2 = x^2 \quad \text{and} \quad b^2 = 25
\]
So, \( a = x \) and \( b = 5 \).
- Apply the formula:
\[
x^2 - 25 = (x - 5)(x + 5)
\]

#### Problem 5: \( x^2 - 36 \)
- Identify \( a \) and \( b \):
\[
a^2 = x^2 \quad \text{and} \quad b^2 = 36
\]
So, \( a = x \) and \( b = 6 \).
- Apply the formula:
\[
x^2 - 36 = (x - 6)(x + 6)
\]

#### Problem 6: \( x^2 - 49 \)
- Identify \( a \) and \( b \):
\[
a^2 = x^2 \quad \text{and} \quad b^2 = 49
\]
So, \( a = x \) and \( b = 7 \).
- Apply the formula:
\[
x^2 - 49 = (x - 7)(x + 7)
\]

#### Problem 7: \( x^2 - 64 \)
- Identify \( a \) and \( b \):
\[
a^2 = x^2 \quad \text{and} \quad b^2 = 64
\]
So, \( a = x \) and \( b = 8 \).
- Apply the formula:
\[
x^2 - 64 = (x - 8)(x + 8)
\]

#### Problem 8: \( x^2 - 81 \)
- Identify \( a \) and \( b \):
\[
a^2 = x^2 \quad \text{and} \quad b^2 = 81
\]
So, \( a = x \) and \( b = 9 \).
- Apply the formula:
\[
x^2 - 81 = (x - 9)(x + 9)
\]

#### Problem 9: \( x^2 - 100 \)
- Identify \( a \) and \( b \):
\[
a^2 = x^2 \quad \text{and} \quad b^2 = 100
\]
So, \( a = x \) and \( b = 10 \).
- Apply the formula:
\[
x^2 - 100 = (x - 10)(x + 10)
\]

#### Problem 10: \( x^2 - 121 \)
- Identify \( a \) and \( b \):
\[
a^2 = x^2 \quad \text{and} \quad b^2 = 121
\]
So, \( a = x \) and \( b = 11 \).
- Apply the formula:
\[
x^2 - 121 = (x - 11)(x + 11)
\]

#### Problem 11: \( x^2 - 144 \)
- Identify \( a \) and \( b \):
\[
a^2 = x^2 \quad \text{and} \quad b^2 = 144
\]
So, \( a = x \) and \( b = 12 \).
- Apply the formula:
\[
x^2 - 144 = (x - 12)(x + 12)
\]

#### Problem 12: \( x^2 - 169 \)
- Identify \( a \) and \( b \):
\[
a^2 = x^2 \quad \text{and} \quad b^2 = 169
\]
So, \( a = x \) and \( b = 13 \).
- Apply the formula:
\[
x^2 - 169 = (x - 13)(x + 13)
\]

---

Section 2: More Complex Differences of Squares

#### Problem 13: \( 25x^2 - y^4 \)
- Identify \( a \) and \( b \):
\[
a^2 = 25x^2 \quad \text{and} \quad b^2 = y^4
\]
So, \( a = 5x \) and \( b = y^2 \).
- Apply the formula:
\[
25x^2 - y^4 = (5x - y^2)(5x + y^2)
\]

#### Problem 14: \( 16x^2 - 9y^2 \)
- Identify \( a \) and \( b \):
\[
a^2 = 16x^2 \quad \text{and} \quad b^2 = 9y^2
\]
So, \( a = 4x \) and \( b = 3y \).
- Apply the formula:
\[
16x^2 - 9y^2 = (4x - 3y)(4x + 3y)
\]

#### Problem 15: \( 36x^2 - 49y^2 \)
- Identify \( a \) and \( b \):
\[
a^2 = 36x^2 \quad \text{and} \quad b^2 = 49y^2
\]
So, \( a = 6x \) and \( b = 7y \).
- Apply the formula:
\[
36x^2 - 49y^2 = (6x - 7y)(6x + 7y)
\]

#### Problem 16: \( 64x^2 - 81y^2 \)
- Identify \( a \) and \( b \):
\[
a^2 = 64x^2 \quad \text{and} \quad b^2 = 81y^2
\]
So, \( a = 8x \) and \( b = 9y \).
- Apply the formula:
\[
64x^2 - 81y^2 = (8x - 9y)(8x + 9y)
\]

#### Problem 17: \( 100x^2 - 121y^2 \)
- Identify \( a \) and \( b \):
\[
a^2 = 100x^2 \quad \text{and} \quad b^2 = 121y^2
\]
So, \( a = 10x \) and \( b = 11y \).
- Apply the formula:
\[
100x^2 - 121y^2 = (10x - 11y)(10x + 11y)
\]

#### Problem 18: \( 144x^2 - 169y^2 \)
- Identify \( a \) and \( b \):
\[
a^2 = 144x^2 \quad \text{and} \quad b^2 = 169y^2
\]
So, \( a = 12x \) and \( b = 13y \).
- Apply the formula:
\[
144x^2 - 169y^2 = (12x - 13y)(12x + 13y)
\]

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Final Answers

\[
\boxed{
\begin{aligned}
1. & \ (x - 2)(x + 2) \\
2. & \ (x - 3)(x + 3) \\
3. & \ (x - 4)(x + 4) \\
4. & \ (x - 5)(x + 5) \\
5. & \ (x - 6)(x + 6) \\
6. & \ (x - 7)(x + 7) \\
7. & \ (x - 8)(x + 8) \\
8. & \ (x - 9)(x + 9) \\
9. & \ (x - 10)(x + 10) \\
10. & \ (x - 11)(x + 11) \\
11. & \ (x - 12)(x + 12) \\
12. & \ (x - 13)(x + 13) \\
13. & \ (5x - y^2)(5x + y^2) \\
14. & \ (4x - 3y)(4x + 3y) \\
15. & \ (6x - 7y)(6x + 7y) \\
16. & \ (8x - 9y)(8x + 9y) \\
17. & \ (10x - 11y)(10x + 11y) \\
18. & \ (12x - 13y)(12x + 13y)
\end{aligned}
}
\]