Factoring the Difference of Two Squares Worksheet | PDF - Free Printable

Let's solve the worksheet titled "Factoring the Difference of Two Squares". The key identity used here is:

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

We will apply this identity to each expression.

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1. $ x^2 - 25 $

- $ 25 = 5^2 $
- So: $ x^2 - 5^2 = (x + 5)(x - 5) $

✔ Answer: $ (x + 5)(x - 5) $

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2. $ a^2 - 9 $

- $ 9 = 3^2 $
- $ a^2 - 3^2 = (a + 3)(a - 3) $

✔ Answer: $ (a + 3)(a - 3) $

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3. $ 9a^2 - 1 $

- $ 9a^2 = (3a)^2 $, $ 1 = 1^2 $
- $ (3a)^2 - 1^2 = (3a + 1)(3a - 1) $

✔ Answer: $ (3a + 1)(3a - 1) $

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4. $ 36b^2 - 1 $

- $ 36b^2 = (6b)^2 $, $ 1 = 1^2 $
- $ (6b)^2 - 1^2 = (6b + 1)(6b - 1) $

✔ Answer: $ (6b + 1)(6b - 1) $

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5. $ 4c^2 - 49 $

- $ 4c^2 = (2c)^2 $, $ 49 = 7^2 $
- $ (2c)^2 - 7^2 = (2c + 7)(2c - 7) $

✔ Answer: $ (2c + 7)(2c - 7) $

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6. $ d^2 - 36 $

- $ 36 = 6^2 $
- $ d^2 - 6^2 = (d + 6)(d - 6) $

✔ Answer: $ (d + 6)(d - 6) $

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7. $ 16e^2 - 25 $

- $ 16e^2 = (4e)^2 $, $ 25 = 5^2 $
- $ (4e)^2 - 5^2 = (4e + 5)(4e - 5) $

✔ Answer: $ (4e + 5)(4e - 5) $

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8. $ 25m^2 - 4 $

- $ 25m^2 = (5m)^2 $, $ 4 = 2^2 $
- $ (5m)^2 - 2^2 = (5m + 2)(5m - 2) $

✔ Answer: $ (5m + 2)(5m - 2) $

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9. $ 9n^2 - 100 $

- $ 9n^2 = (3n)^2 $, $ 100 = 10^2 $
- $ (3n)^2 - 10^2 = (3n + 10)(3n - 10) $

✔ Answer: $ (3n + 10)(3n - 10) $

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10. $ 49p^2 - 81 $

- $ 49p^2 = (7p)^2 $, $ 81 = 9^2 $
- $ (7p)^2 - 9^2 = (7p + 9)(7p - 9) $

✔ Answer: $ (7p + 9)(7p - 9) $

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11. $ 4x^2 - 9 $

- $ 4x^2 = (2x)^2 $, $ 9 = 3^2 $
- $ (2x)^2 - 3^2 = (2x + 3)(2x - 3) $

✔ Answer: $ (2x + 3)(2x - 3) $

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12. $ 25y^2 - z^2 $

- $ 25y^2 = (5y)^2 $, $ z^2 = z^2 $
- $ (5y)^2 - z^2 = (5y + z)(5y - z) $

✔ Answer: $ (5y + z)(5y - z) $

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13. $ a^2 - 25b^2 $

- $ a^2 = a^2 $, $ 25b^2 = (5b)^2 $
- $ a^2 - (5b)^2 = (a + 5b)(a - 5b) $

✔ Answer: $ (a + 5b)(a - 5b) $

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14. $ 16m^2 - 81n^2 $

- $ 16m^2 = (4m)^2 $, $ 81n^2 = (9n)^2 $
- $ (4m)^2 - (9n)^2 = (4m + 9n)(4m - 9n) $

✔ Answer: $ (4m + 9n)(4m - 9n) $

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15. $ 25x^2 - 16y^2 $

- $ 25x^2 = (5x)^2 $, $ 16y^2 = (4y)^2 $
- $ (5x)^2 - (4y)^2 = (5x + 4y)(5x - 4y) $

✔ Answer: $ (5x + 4y)(5x - 4y) $

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16. $ 36p^2 - 9q^2 $

- First factor out common factor: $ 9(4p^2 - q^2) $
- $ 4p^2 = (2p)^2 $, $ q^2 = q^2 $
- $ 4p^2 - q^2 = (2p + q)(2p - q) $
- So: $ 9(2p + q)(2p - q) $

✔ Answer: $ 9(2p + q)(2p - q) $

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17. $ x^{10} - 4 $

- $ x^{10} = (x^5)^2 $, $ 4 = 2^2 $
- $ (x^5)^2 - 2^2 = (x^5 + 2)(x^5 - 2) $

✔ Answer: $ (x^5 + 2)(x^5 - 2) $

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18. $ 2x^2 - 18 $

- Factor out common factor: $ 2(x^2 - 9) $
- $ x^2 - 9 = x^2 - 3^2 = (x + 3)(x - 3) $
- So: $ 2(x + 3)(x - 3) $

✔ Answer: $ 2(x + 3)(x - 3) $

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19. $ 3x^2 - 75 $

- Factor out common factor: $ 3(x^2 - 25) $
- $ x^2 - 25 = (x + 5)(x - 5) $
- So: $ 3(x + 5)(x - 5) $

✔ Answer: $ 3(x + 5)(x - 5) $

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20. $ 6y^2 - 54 $

- Factor out common factor: $ 6(y^2 - 9) $
- $ y^2 - 9 = (y + 3)(y - 3) $
- So: $ 6(y + 3)(y - 3) $

✔ Answer: $ 6(y + 3)(y - 3) $

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✔ Final Answers Summary:

| Problem | Factored Form |
|--------|----------------|
| 1 | $ (x + 5)(x - 5) $ |
| 2 | $ (a + 3)(a - 3) $ |
| 3 | $ (3a + 1)(3a - 1) $ |
| 4 | $ (6b + 1)(6b - 1) $ |
| 5 | $ (2c + 7)(2c - 7) $ |
| 6 | $ (d + 6)(d - 6) $ |
| 7 | $ (4e + 5)(4e - 5) $ |
| 8 | $ (5m + 2)(5m - 2) $ |
| 9 | $ (3n + 10)(3n - 10) $ |
| 10 | $ (7p + 9)(7p - 9) $ |
| 11 | $ (2x + 3)(2x - 3) $ |
| 12 | $ (5y + z)(5y - z) $ |
| 13 | $ (a + 5b)(a - 5b) $ |
| 14 | $ (4m + 9n)(4m - 9n) $ |
| 15 | $ (5x + 4y)(5x - 4y) $ |
| 16 | $ 9(2p + q)(2p - q) $ |
| 17 | $ (x^5 + 2)(x^5 - 2) $ |
| 18 | $ 2(x + 3)(x - 3) $ |
| 19 | $ 3(x + 5)(x - 5) $ |
| 20 | $ 6(y + 3)(y - 3) $ |

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🔍 Key Tips:

- Always check if you can factor out a GCF first.
- Recognize perfect squares: $ 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, \dots $
- Remember that $ a^2 - b^2 = (a + b)(a - b) $
- For higher powers like $ x^{10} $, treat as $ (x^5)^2 $

Let me know if you'd like this in printable format or need help with any specific problem!