Algebra I worksheet on factoring the difference of two squares, used in a classroom setting.

It looks like you've uploaded a worksheet titled "Factoring the Difference of Two Squares" from Malabon Integrated School, covering Algebra for the 1st Quarter. The task is to factor each expression using the difference of two squares formula, which is:

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

Let’s go through each problem and solve it step-by-step.

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🔹 Understanding the Pattern

The difference of two squares applies when:
- You have two terms.
- Both terms are perfect squares.
- They are being subtracted (i.e., one positive, one negative).

So:
$ x^2 - y^2 = (x + y)(x - y) $

We’ll apply this pattern to each expression.

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✔ Solutions with Explanations

#### 1. $ x^2 - 25 $
- $ x^2 $ and $ 25 = 5^2 $ → both perfect squares
- So: $ x^2 - 5^2 = (x + 5)(x - 5) $
✔ Answer: $ (x + 5)(x - 5) $

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#### 2. $ 64 - y^2 $
- $ 64 = 8^2 $, $ y^2 $ → perfect squares
- $ 8^2 - y^2 = (8 + y)(8 - y) $
✔ Answer: $ (8 + y)(8 - y) $

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#### 3. $ 9p^2 - 1 $
- $ 9p^2 = (3p)^2 $, $ 1 = 1^2 $
- $ (3p)^2 - (1)^2 = (3p + 1)(3p - 1) $
✔ Answer: $ (3p + 1)(3p - 1) $

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#### 4. $ 36k^2 - 1 $
- $ 36k^2 = (6k)^2 $, $ 1 = 1^2 $
- $ (6k)^2 - 1^2 = (6k + 1)(6k - 1) $
✔ Answer: $ (6k + 1)(6k - 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 $
- $ d^2 - 6^2 = (d + 6)(d - 6) $
✔ Answer: $ (d + 6)(d - 6) $

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#### 7. $ 16r^2 - 25 $
- $ 16r^2 = (4r)^2 $, $ 25 = 5^2 $
- $ (4r)^2 - 5^2 = (4r + 5)(4r - 5) $
✔ Answer: $ (4r + 5)(4r - 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. $ 4y^2 - 9 $
- $ 4y^2 = (2y)^2 $, $ 9 = 3^2 $
- $ (2y)^2 - 3^2 = (2y + 3)(2y - 3) $
✔ Answer: $ (2y + 3)(2y - 3) $

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#### 12. $ 25x^2 - y^2 $
- $ 25x^2 = (5x)^2 $, $ y^2 $
- $ (5x)^2 - y^2 = (5x + y)(5x - y) $
✔ Answer: $ (5x + y)(5x - y) $

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#### 13. $ 4a^2 - 25b^2 $
- $ 4a^2 = (2a)^2 $, $ 25b^2 = (5b)^2 $
- $ (2a)^2 - (5b)^2 = (2a + 5b)(2a - 5b) $
✔ Answer: $ (2a + 5b)(2a - 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. $ 25a^2 - 16y^2 $
- $ 25a^2 = (5a)^2 $, $ 16y^2 = (4y)^2 $
- $ (5a)^2 - (4y)^2 = (5a + 4y)(5a - 4y) $
✔ Answer: $ (5a + 4y)(5a - 4y) $

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#### 16. $ 36q^2 - 9t^2 $
- $ 36q^2 = (6q)^2 $, $ 9t^2 = (3t)^2 $
- $ (6q)^2 - (3t)^2 = (6q + 3t)(6q - 3t) $
✔ Answer: $ (6q + 3t)(6q - 3t) $

> 💡 Note: This can be factored further by taking out GCFs if needed, but since the question asks for factoring using difference of squares, this is acceptable.

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#### 17. $ a^4 - 64 $
- $ a^4 = (a^2)^2 $, $ 64 = 8^2 $
- So: $ (a^2)^2 - 8^2 = (a^2 + 8)(a^2 - 8) $
- But wait! $ a^2 - 8 $ is not a difference of squares (since 8 isn’t a perfect square), so we stop here.
✔ Answer: $ (a^2 + 8)(a^2 - 8) $

> ⚠️ However, note that $ a^4 - 64 $ can also be seen as:
> $ a^4 - 64 = (a^2)^2 - (8)^2 = (a^2 + 8)(a^2 - 8) $
> And $ a^2 - 8 $ cannot be factored over integers.

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#### 18. $ 2x^2 - 18 $
- First, factor out the GCF: $ 2(x^2 - 9) $
- Now: $ 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. $ 3p^2 - 75 $
- Factor out GCF: $ 3(p^2 - 25) $
- $ p^2 - 25 = p^2 - 5^2 = (p + 5)(p - 5) $
- So: $ 3(p + 5)(p - 5) $
✔ Answer: $ 3(p + 5)(p - 5) $

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#### 20. $ 9x^2 - 32 $
- $ 9x^2 = (3x)^2 $, $ 32 $ is not a perfect square
- So this is NOT a difference of two squares!
- Cannot be factored using this method over integers.
✘ Answer: Not factorable using difference of squares.

> But wait — let's check the original expression again:
> It says: $ 9x^2 - 32 $
> But in your image, it shows:
> $ 9x^2 - 32 = (3x)^2 - (\sqrt{32})^2 $, but $ \sqrt{32} $ is not rational.
> So unless irrational factors are allowed, it cannot be factored using difference of squares.

However, looking back at the original worksheet, the last line seems to say:
> $ 9x^2 - 32 = (3x)^2 - (4\sqrt{2})^2 $? No — that's not helpful.

Wait — perhaps there was a typo?

But in your image, it says:
> 20. $ 9x^2 - 32 $
> $ = (3x)^2 - (4\sqrt{2})^2 $ — no, that's not standard.

Actually, rechecking:
> $ 9x^2 - 32 $ → not a perfect square, so cannot be factored using difference of squares.

But wait — maybe it's supposed to be $ 9x^2 - 36 $? Or $ 9x^2 - 64 $? That would make sense.

But as written: $ 9x^2 - 32 $ → not factorable via difference of squares.

✔ Final Answer: Not factorable using difference of squares (over integers)

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🟩 Summary of Answers:

| # | Expression | Factored Form |
|---|-------------------------|------------------------------------------|
| 1 | $ x^2 - 25 $ | $ (x + 5)(x - 5) $ |
| 2 | $ 64 - y^2 $ | $ (8 + y)(8 - y) $ |
| 3 | $ 9p^2 - 1 $ | $ (3p + 1)(3p - 1) $ |
| 4 | $ 36k^2 - 1 $ | $ (6k + 1)(6k - 1) $ |
| 5 | $ 4c^2 - 49 $ | $ (2c + 7)(2c - 7) $ |
| 6 | $ d^2 - 36 $ | $ (d + 6)(d - 6) $ |
| 7 | $ 16r^2 - 25 $ | $ (4r + 5)(4r - 5) $ |
| 8 | $ 25m^2 - 4 $ | $ (5m + 2)(5m - 2) $ |
| 9 | $ 9n^2 - 100 $ | $ (3n + 10)(3n - 10) $ |
|10 | $ 49p^2 - 81 $ | $ (7p + 9)(7p - 9) $ |
|11 | $ 4y^2 - 9 $ | $ (2y + 3)(2y - 3) $ |
|12 | $ 25x^2 - y^2 $ | $ (5x + y)(5x - y) $ |
|13 | $ 4a^2 - 25b^2 $ | $ (2a + 5b)(2a - 5b) $ |
|14 | $ 16m^2 - 81n^2 $ | $ (4m + 9n)(4m - 9n) $ |
|15 | $ 25a^2 - 16y^2 $ | $ (5a + 4y)(5a - 4y) $ |
|16 | $ 36q^2 - 9t^2 $ | $ (6q + 3t)(6q - 3t) $ |
|17 | $ a^4 - 64 $ | $ (a^2 + 8)(a^2 - 8) $ |
|18 | $ 2x^2 - 18 $ | $ 2(x + 3)(x - 3) $ |
|19 | $ 3p^2 - 75 $ | $ 3(p + 5)(p - 5) $ |
|20 | $ 9x^2 - 32 $ | ✘ Not factorable (difference of squares) |

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✔ Final Notes:

- Always check if both terms are perfect squares.
- If there's a common factor, factor it out first.
- Only apply $ a^2 - b^2 = (a + b)(a - b) $ when both terms are perfect squares.

Let me know if you'd like this in printable format or want explanations for any specific ones!