You're asked to
factor each expression completely. All of these are examples of the
difference of squares, which follows the formula:
>
a² – b² = (a + b)(a – b)
We’ll apply this formula to each problem.
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1. $ y^2 - 1 $
- This is $ y^2 - 1^2 $
- So, $ a = y $, $ b = 1 $
- Factored: $ \boxed{(y + 1)(y - 1)} $
---
2. $ 9u^2 - 4 $
- $ 9u^2 = (3u)^2 $, $ 4 = 2^2 $
- So, $ a = 3u $, $ b = 2 $
- Factored: $ \boxed{(3u + 2)(3u - 2)} $
---
3. $ 144x^2 - 25 $
- $ 144x^2 = (12x)^2 $, $ 25 = 5^2 $
- So, $ a = 12x $, $ b = 5 $
- Factored: $ \boxed{(12x + 5)(12x - 5)} $
---
4. $ 121 - 9y^2 $
- $ 121 = 11^2 $, $ 9y^2 = (3y)^2 $
- So, $ a = 11 $, $ b = 3y $
- Factored: $ \boxed{(11 + 3y)(11 - 3y)} $
*(Note: Order doesn’t matter — you can also write it as $ (11 - 3y)(11 + 3y) $)*
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5. $ 81x^2 - 121 $
- $ 81x^2 = (9x)^2 $, $ 121 = 11^2 $
- So, $ a = 9x $, $ b = 11 $
- Factored: $ \boxed{(9x + 11)(9x - 11)} $
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6. $ 121y^2 - 36 $
- $ 121y^2 = (11y)^2 $, $ 36 = 6^2 $
- So, $ a = 11y $, $ b = 6 $
- Factored: $ \boxed{(11y + 6)(11y - 6)} $
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7. $ 64a^2 - 25 $
- $ 64a^2 = (8a)^2 $, $ 25 = 5^2 $
- So, $ a = 8a $, $ b = 5 $
- Factored: $ \boxed{(8a + 5)(8a - 5)} $
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8. $ 25u^2 - 1 $
- $ 25u^2 = (5u)^2 $, $ 1 = 1^2 $
- So, $ a = 5u $, $ b = 1 $
- Factored: $ \boxed{(5u + 1)(5u - 1)} $
---
9. $ 49x^2 - 4 $
- $ 49x^2 = (7x)^2 $, $ 4 = 2^2 $
- So, $ a = 7x $, $ b = 2 $
- Factored: $ \boxed{(7x + 2)(7x - 2)} $
---
10. $ 81 - 36x^2 $
- $ 81 = 9^2 $, $ 36x^2 = (6x)^2 $
- So, $ a = 9 $, $ b = 6x $
- Factored: $ \boxed{(9 + 6x)(9 - 6x)} $
> 💡 *Optional simplification*: You could factor out a common factor from each binomial:
>
> $ (9 + 6x) = 3(3 + 2x) $
> $ (9 - 6x) = 3(3 - 2x) $
> So overall: $ 3(3 + 2x) \cdot 3(3 - 2x) = 9(3 + 2x)(3 - 2x) $
>
> But since the instruction says “factor completely” and doesn't specify factoring out GCFs first, the standard difference of squares form is acceptable. However, if your teacher expects fully factored form including GCF, then:
>
>
Fully factored: $ \boxed{9(3 + 2x)(3 - 2x)} $
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##
✔ Final Answers:
1. $ (y + 1)(y - 1) $
2. $ (3u + 2)(3u - 2) $
3. $ (12x + 5)(12x - 5) $
4. $ (11 + 3y)(11 - 3y) $
5. $ (9x + 11)(9x - 11) $
6. $ (11y + 6)(11y - 6) $
7. $ (8a + 5)(8a - 5) $
8. $ (5u + 1)(5u - 1) $
9. $ (7x + 2)(7x - 2) $
10. $ \boxed{(9 + 6x)(9 - 6x)} $ or fully simplified: $ \boxed{9(3 + 2x)(3 - 2x)} $
Let me know if you want to see step-by-step work for any one in more detail!
Parent Tip: Review the logic above to help your child master the concept of factoring difference of two squares worksheet.
