Seventy-Two Factoring Difference of Squares Problems Worksheet for ... - Free Printable
Educational worksheet: Seventy-Two Factoring Difference of Squares Problems Worksheet for .... Download and print for classroom or home learning activities.
JPG
300×225
11.2 KB
Free · Personal Use
Quality Assured by Worksheets Library Team
Reviewed for educational accuracy and age-appropriateness
ID: #1391732
⭐
Show Answer Key & Explanations
Step-by-step solution for: Seventy-Two Factoring Difference of Squares Problems Worksheet for ...
▼
Show Answer Key & Explanations
Step-by-step solution for: Seventy-Two Factoring Difference of Squares Problems Worksheet for ...
It looks like you've uploaded an image containing a worksheet on factoring the difference of squares. Since I can't view images directly, I’ll help you solve the problems based on the text you provided.
The difference of squares formula is:
$$
a^2 - b^2 = (a - b)(a + b)
$$
We'll go through each problem step by step.
---
#### Part 1: Basic Difference of Squares
These are straightforward expressions that match $ a^2 - b^2 $.
1. $ x^2 - 4 $
$ = x^2 - 2^2 = (x - 2)(x + 2) $
2. $ x^2 - 9 $
$ = x^2 - 3^2 = (x - 3)(x + 3) $
3. $ x^2 - 16 $
$ = x^2 - 4^2 = (x - 4)(x + 4) $
4. $ x^2 - 36 $
$ = x^2 - 6^2 = (x - 6)(x + 6) $
5. $ x^2 - 49 $
$ = x^2 - 7^2 = (x - 7)(x + 7) $
6. $ x^2 - 64 $
$ = x^2 - 8^2 = (x - 8)(x + 8) $
7. $ x^2 - 100 $
$ = x^2 - 10^2 = (x - 10)(x + 10) $
8. $ x^2 - 121 $
$ = x^2 - 11^2 = (x - 11)(x + 11) $
9. $ x^2 - 144 $
$ = x^2 - 12^2 = (x - 12)(x + 12) $
10. $ x^2 - 169 $
$ = x^2 - 13^2 = (x - 13)(x + 13) $
11. $ x^2 - 196 $
$ = x^2 - 14^2 = (x - 14)(x + 14) $
12. $ x^2 - 225 $
$ = x^2 - 15^2 = (x - 15)(x + 15) $
13. $ x^2 - 256 $
$ = x^2 - 16^2 = (x - 16)(x + 16) $
14. $ x^2 - 289 $
$ = x^2 - 17^2 = (x - 17)(x + 17) $
15. $ x^2 - 324 $
$ = x^2 - 18^2 = (x - 18)(x + 18) $
16. $ x^2 - 361 $
$ = x^2 - 19^2 = (x - 19)(x + 19) $
17. $ x^2 - 400 $
$ = x^2 - 20^2 = (x - 20)(x + 20) $
18. $ x^2 - 441 $
$ = x^2 - 21^2 = (x - 21)(x + 21) $
19. $ x^2 - 484 $
$ = x^2 - 22^2 = (x - 22)(x + 22) $
20. $ x^2 - 529 $
$ = x^2 - 23^2 = (x - 23)(x + 23) $
21. $ x^2 - 576 $
$ = x^2 - 24^2 = (x - 24)(x + 24) $
22. $ x^2 - 625 $
$ = x^2 - 25^2 = (x - 25)(x + 25) $
23. $ x^2 - y^2 $
$ = (x - y)(x + y) $
24. $ x^2 - z^2 $
$ = (x - z)(x + z) $
25. $ x^2 - 4 $
$ = x^2 - 2^2 = (x - 2)(x + 2) $
26. $ x^2 - x^2 $? Wait — this is $ x^2 - x^2 = 0 $. But likely a typo. Let's assume it's meant to be $ x^2 - 1 $ or $ x^2 - y^2 $. But as written:
$ x^2 - x^2 = 0 $ → factors as $ 0 $, but not useful. Possibly a mistake.
27. $ 4x^2 - 9 $
$ = (2x)^2 - 3^2 = (2x - 3)(2x + 3) $
28. $ 4x^2 - 36 $
$ = 4(x^2 - 9) = 4(x - 3)(x + 3) $
29. $ 25x^2 - 36 $
$ = (5x)^2 - 6^2 = (5x - 6)(5x + 6) $
30. $ 25x^2 - 441 $
$ = (5x)^2 - 21^2 = (5x - 21)(5x + 21) $
31. $ 81x^2 - 225 $
$ = (9x)^2 - 15^2 = (9x - 15)(9x + 15) $
You can factor out common factor: $ = 9(9x - 15)(9x + 15) $? No — better to simplify:
Actually: $ 81x^2 - 225 = 9(9x^2 - 25) = 9((3x)^2 - 5^2) = 9(3x - 5)(3x + 5) $
32. $ 100x^2 - 144 $
$ = (10x)^2 - 12^2 = (10x - 12)(10x + 12) $
Factor out GCF: $ = 4(25x^2 - 36) = 4((5x)^2 - 6^2) = 4(5x - 6)(5x + 6) $
33. $ x^2 - 169 $
$ = x^2 - 13^2 = (x - 13)(x + 13) $
34. $ 4x^2 - 169 $
$ = (2x)^2 - 13^2 = (2x - 13)(2x + 13) $
35. $ 4x^2 - 169y^2 $
$ = (2x)^2 - (13y)^2 = (2x - 13y)(2x + 13y) $
36. $ 25x^2 - 49y^2 $
$ = (5x)^2 - (7y)^2 = (5x - 7y)(5x + 7y) $
---
#### Part 2: More Complex Expressions
Now let’s solve the second set:
1. $ 49 - x^2 $
$ = 7^2 - x^2 = (7 - x)(7 + x) $
Or: $ -(x^2 - 49) = -(x - 7)(x + 7) $, but standard form is $ (7 - x)(7 + x) $
2. $ 3x^2 - 75 $
First factor out GCF: $ = 3(x^2 - 25) = 3(x - 5)(x + 5) $
3. $ 25x^2 - 4 $
$ = (5x)^2 - 2^2 = (5x - 2)(5x + 2) $
4. $ 121x^2 - 81 $
$ = (11x)^2 - 9^2 = (11x - 9)(11x + 9) $
5. $ 256 - x^2 $
$ = 16^2 - x^2 = (16 - x)(16 + x) $
6. $ 64 - 54 $
Wait — this is $ 64 - 54 = 10 $, not a binomial. Likely a typo. Maybe it's $ 64 - x^2 $? Or $ 64 - 9x^2 $?
If it's $ 64 - 9x^2 $, then:
$ = 8^2 - (3x)^2 = (8 - 3x)(8 + 3x) $
But as written: $ 64 - 54 = 10 $ — not factorable as difference of squares.
7. $ 25x^2 - y^2 $
$ = (5x)^2 - y^2 = (5x - y)(5x + y) $
---
Here are the correct factorizations:
#### Part 1:
1. $ (x - 2)(x + 2) $
2. $ (x - 3)(x + 3) $
3. $ (x - 4)(x + 4) $
4. $ (x - 6)(x + 6) $
5. $ (x - 7)(x + 7) $
6. $ (x - 8)(x + 8) $
7. $ (x - 10)(x + 10) $
8. $ (x - 11)(x + 11) $
9. $ (x - 12)(x + 12) $
10. $ (x - 13)(x + 13) $
11. $ (x - 14)(x + 14) $
12. $ (x - 15)(x + 15) $
13. $ (x - 16)(x + 16) $
14. $ (x - 17)(x + 17) $
15. $ (x - 18)(x + 18) $
16. $ (x - 19)(x + 19) $
17. $ (x - 20)(x + 20) $
18. $ (x - 21)(x + 21) $
19. $ (x - 22)(x + 22) $
20. $ (x - 23)(x + 23) $
21. $ (x - 24)(x + 24) $
22. $ (x - 25)(x + 25) $
23. $ (x - y)(x + y) $
24. $ (x - z)(x + z) $
25. $ (x - 2)(x + 2) $
26. $ x^2 - x^2 = 0 $ → $ 0 $ (not factorable in meaningful way)
27. $ (2x - 3)(2x + 3) $
28. $ 4(x - 3)(x + 3) $
29. $ (5x - 6)(5x + 6) $
30. $ (5x - 21)(5x + 21) $
31. $ 9(3x - 5)(3x + 5) $
32. $ 4(5x - 6)(5x + 6) $
33. $ (x - 13)(x + 13) $
34. $ (2x - 13)(2x + 13) $
35. $ (2x - 13y)(2x + 13y) $
36. $ (5x - 7y)(5x + 7y) $
#### Part 2:
1. $ (7 - x)(7 + x) $ or $ -(x - 7)(x + 7) $
2. $ 3(x - 5)(x + 5) $
3. $ (5x - 2)(5x + 2) $
4. $ (11x - 9)(11x + 9) $
5. $ (16 - x)(16 + x) $
6. $ 64 - 54 = 10 $ → Not a difference of squares; possibly typo. If $ 64 - 9x^2 $, then $ (8 - 3x)(8 + 3x) $
7. $ (5x - y)(5x + y) $
---
- Always look for perfect squares.
- Factor out GCF first if needed.
- Use the identity: $ a^2 - b^2 = (a - b)(a + b) $
- For expressions like $ 4x^2 - 9 $, think $ (2x)^2 - 3^2 $
Let me know if you'd like these formatted neatly or if any problem was misread!
The difference of squares formula is:
$$
a^2 - b^2 = (a - b)(a + b)
$$
We'll go through each problem step by step.
---
Factoring Difference of Squares
#### Part 1: Basic Difference of Squares
These are straightforward expressions that match $ a^2 - b^2 $.
1. $ x^2 - 4 $
$ = x^2 - 2^2 = (x - 2)(x + 2) $
2. $ x^2 - 9 $
$ = x^2 - 3^2 = (x - 3)(x + 3) $
3. $ x^2 - 16 $
$ = x^2 - 4^2 = (x - 4)(x + 4) $
4. $ x^2 - 36 $
$ = x^2 - 6^2 = (x - 6)(x + 6) $
5. $ x^2 - 49 $
$ = x^2 - 7^2 = (x - 7)(x + 7) $
6. $ x^2 - 64 $
$ = x^2 - 8^2 = (x - 8)(x + 8) $
7. $ x^2 - 100 $
$ = x^2 - 10^2 = (x - 10)(x + 10) $
8. $ x^2 - 121 $
$ = x^2 - 11^2 = (x - 11)(x + 11) $
9. $ x^2 - 144 $
$ = x^2 - 12^2 = (x - 12)(x + 12) $
10. $ x^2 - 169 $
$ = x^2 - 13^2 = (x - 13)(x + 13) $
11. $ x^2 - 196 $
$ = x^2 - 14^2 = (x - 14)(x + 14) $
12. $ x^2 - 225 $
$ = x^2 - 15^2 = (x - 15)(x + 15) $
13. $ x^2 - 256 $
$ = x^2 - 16^2 = (x - 16)(x + 16) $
14. $ x^2 - 289 $
$ = x^2 - 17^2 = (x - 17)(x + 17) $
15. $ x^2 - 324 $
$ = x^2 - 18^2 = (x - 18)(x + 18) $
16. $ x^2 - 361 $
$ = x^2 - 19^2 = (x - 19)(x + 19) $
17. $ x^2 - 400 $
$ = x^2 - 20^2 = (x - 20)(x + 20) $
18. $ x^2 - 441 $
$ = x^2 - 21^2 = (x - 21)(x + 21) $
19. $ x^2 - 484 $
$ = x^2 - 22^2 = (x - 22)(x + 22) $
20. $ x^2 - 529 $
$ = x^2 - 23^2 = (x - 23)(x + 23) $
21. $ x^2 - 576 $
$ = x^2 - 24^2 = (x - 24)(x + 24) $
22. $ x^2 - 625 $
$ = x^2 - 25^2 = (x - 25)(x + 25) $
23. $ x^2 - y^2 $
$ = (x - y)(x + y) $
24. $ x^2 - z^2 $
$ = (x - z)(x + z) $
25. $ x^2 - 4 $
$ = x^2 - 2^2 = (x - 2)(x + 2) $
26. $ x^2 - x^2 $? Wait — this is $ x^2 - x^2 = 0 $. But likely a typo. Let's assume it's meant to be $ x^2 - 1 $ or $ x^2 - y^2 $. But as written:
$ x^2 - x^2 = 0 $ → factors as $ 0 $, but not useful. Possibly a mistake.
27. $ 4x^2 - 9 $
$ = (2x)^2 - 3^2 = (2x - 3)(2x + 3) $
28. $ 4x^2 - 36 $
$ = 4(x^2 - 9) = 4(x - 3)(x + 3) $
29. $ 25x^2 - 36 $
$ = (5x)^2 - 6^2 = (5x - 6)(5x + 6) $
30. $ 25x^2 - 441 $
$ = (5x)^2 - 21^2 = (5x - 21)(5x + 21) $
31. $ 81x^2 - 225 $
$ = (9x)^2 - 15^2 = (9x - 15)(9x + 15) $
You can factor out common factor: $ = 9(9x - 15)(9x + 15) $? No — better to simplify:
Actually: $ 81x^2 - 225 = 9(9x^2 - 25) = 9((3x)^2 - 5^2) = 9(3x - 5)(3x + 5) $
32. $ 100x^2 - 144 $
$ = (10x)^2 - 12^2 = (10x - 12)(10x + 12) $
Factor out GCF: $ = 4(25x^2 - 36) = 4((5x)^2 - 6^2) = 4(5x - 6)(5x + 6) $
33. $ x^2 - 169 $
$ = x^2 - 13^2 = (x - 13)(x + 13) $
34. $ 4x^2 - 169 $
$ = (2x)^2 - 13^2 = (2x - 13)(2x + 13) $
35. $ 4x^2 - 169y^2 $
$ = (2x)^2 - (13y)^2 = (2x - 13y)(2x + 13y) $
36. $ 25x^2 - 49y^2 $
$ = (5x)^2 - (7y)^2 = (5x - 7y)(5x + 7y) $
---
#### Part 2: More Complex Expressions
Now let’s solve the second set:
1. $ 49 - x^2 $
$ = 7^2 - x^2 = (7 - x)(7 + x) $
Or: $ -(x^2 - 49) = -(x - 7)(x + 7) $, but standard form is $ (7 - x)(7 + x) $
2. $ 3x^2 - 75 $
First factor out GCF: $ = 3(x^2 - 25) = 3(x - 5)(x + 5) $
3. $ 25x^2 - 4 $
$ = (5x)^2 - 2^2 = (5x - 2)(5x + 2) $
4. $ 121x^2 - 81 $
$ = (11x)^2 - 9^2 = (11x - 9)(11x + 9) $
5. $ 256 - x^2 $
$ = 16^2 - x^2 = (16 - x)(16 + x) $
6. $ 64 - 54 $
Wait — this is $ 64 - 54 = 10 $, not a binomial. Likely a typo. Maybe it's $ 64 - x^2 $? Or $ 64 - 9x^2 $?
If it's $ 64 - 9x^2 $, then:
$ = 8^2 - (3x)^2 = (8 - 3x)(8 + 3x) $
But as written: $ 64 - 54 = 10 $ — not factorable as difference of squares.
7. $ 25x^2 - y^2 $
$ = (5x)^2 - y^2 = (5x - y)(5x + y) $
---
✔ Final Answers Summary:
Here are the correct factorizations:
#### Part 1:
1. $ (x - 2)(x + 2) $
2. $ (x - 3)(x + 3) $
3. $ (x - 4)(x + 4) $
4. $ (x - 6)(x + 6) $
5. $ (x - 7)(x + 7) $
6. $ (x - 8)(x + 8) $
7. $ (x - 10)(x + 10) $
8. $ (x - 11)(x + 11) $
9. $ (x - 12)(x + 12) $
10. $ (x - 13)(x + 13) $
11. $ (x - 14)(x + 14) $
12. $ (x - 15)(x + 15) $
13. $ (x - 16)(x + 16) $
14. $ (x - 17)(x + 17) $
15. $ (x - 18)(x + 18) $
16. $ (x - 19)(x + 19) $
17. $ (x - 20)(x + 20) $
18. $ (x - 21)(x + 21) $
19. $ (x - 22)(x + 22) $
20. $ (x - 23)(x + 23) $
21. $ (x - 24)(x + 24) $
22. $ (x - 25)(x + 25) $
23. $ (x - y)(x + y) $
24. $ (x - z)(x + z) $
25. $ (x - 2)(x + 2) $
26. $ x^2 - x^2 = 0 $ → $ 0 $ (not factorable in meaningful way)
27. $ (2x - 3)(2x + 3) $
28. $ 4(x - 3)(x + 3) $
29. $ (5x - 6)(5x + 6) $
30. $ (5x - 21)(5x + 21) $
31. $ 9(3x - 5)(3x + 5) $
32. $ 4(5x - 6)(5x + 6) $
33. $ (x - 13)(x + 13) $
34. $ (2x - 13)(2x + 13) $
35. $ (2x - 13y)(2x + 13y) $
36. $ (5x - 7y)(5x + 7y) $
#### Part 2:
1. $ (7 - x)(7 + x) $ or $ -(x - 7)(x + 7) $
2. $ 3(x - 5)(x + 5) $
3. $ (5x - 2)(5x + 2) $
4. $ (11x - 9)(11x + 9) $
5. $ (16 - x)(16 + x) $
6. $ 64 - 54 = 10 $ → Not a difference of squares; possibly typo. If $ 64 - 9x^2 $, then $ (8 - 3x)(8 + 3x) $
7. $ (5x - y)(5x + y) $
---
📌 Key Tips:
- Always look for perfect squares.
- Factor out GCF first if needed.
- Use the identity: $ a^2 - b^2 = (a - b)(a + b) $
- For expressions like $ 4x^2 - 9 $, think $ (2x)^2 - 3^2 $
Let me know if you'd like these formatted neatly or if any problem was misread!
Parent Tip: Review the logic above to help your child master the concept of factoring difference of two squares worksheet.
