Factoring Ax 2 Bx C Worksheet - Fill Online, Printable, Fillable ... - Free Printable

Since I can't view or access images directly, I’ll help you solve the problems from Worksheet 3 (Section 5.3) on Factoring Trinomials of the form $ ax^2 + bx + c $ based on the list you provided.

Let’s go through a few examples step-by-step to demonstrate the method and then provide solutions for all the problems. The general approach is:

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🔧 Method: Factoring $ ax^2 + bx + c $

For trinomials of the form $ ax^2 + bx + c $, we want to factor it into two binomials:
$$
(ax^2 + bx + c) = (px + q)(rx + s)
$$

We look for two numbers that:
- Multiply to $ a \cdot c $
- Add to $ b $

Then use splitting the middle term and factoring by grouping.

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Let’s solve each problem one by one.

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✔ 1. $ 2x^2 + 5x + 3 $

Find two numbers that multiply to $ 2 \cdot 3 = 6 $, add to $ 5 $.
→ $ 2 $ and $ 3 $

Split middle term:
$$
2x^2 + 2x + 3x + 3 = 2x(x + 1) + 3(x + 1) = (2x + 3)(x + 1)
$$

✔ Answer: $ (2x + 3)(x + 1) $

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✔ 2. $ 2x^2 + 5x + 2 $

$ ac = 4 $, need two numbers that multiply to 4, add to 5 → $ 4 $ and $ 1 $

Split:
$$
2x^2 + 4x + x + 2 = 2x(x + 2) + 1(x + 2) = (2x + 1)(x + 2)
$$

✔ Answer: $ (2x + 1)(x + 2) $

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✔ 3. $ 2y^2 - 13y + 20 $

$ ac = 40 $, need two numbers that multiply to 40, add to -13 → $ -8 $, $ -5 $

Split:
$$
2y^2 - 8y - 5y + 20 = 2y(y - 4) -5(y - 4) = (2y - 5)(y - 4)
$$

✔ Answer: $ (2y - 5)(y - 4) $

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✔ 4. $ 2y^2 + 11y + 15 $

$ ac = 30 $, need two numbers that multiply to 30, add to 11 → $ 6 $, $ 5 $

Split:
$$
2y^2 + 6y + 5y + 15 = 2y(y + 3) + 5(y + 3) = (2y + 5)(y + 3)
$$

✔ Answer: $ (2y + 5)(y + 3) $

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✔ 5. $ 2t^2 + 7t - 15 $

$ ac = -30 $, need two numbers that multiply to -30, add to 7 → $ 10 $, $ -3 $

Split:
$$
2t^2 + 10t - 3t - 15 = 2t(t + 5) -3(t + 5) = (2t - 3)(t + 5)
$$

✔ Answer: $ (2t - 3)(t + 5) $

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✔ 6. $ 2t^2 - 9t - 35 $

$ ac = -70 $, need two numbers that multiply to -70, add to -9 → $ -14 $, $ 5 $

Split:
$$
2t^2 -14t + 5t - 35 = 2t(t - 7) + 5(t - 7) = (2t + 5)(t - 7)
$$

✔ Answer: $ (2t + 5)(t - 7) $

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✔ 7. $ 2x^2 + 3x - 20 $

$ ac = -40 $, need two numbers that multiply to -40, add to 3 → $ 8 $, $ -5 $

Split:
$$
2x^2 + 8x - 5x - 20 = 2x(x + 4) -5(x + 4) = (2x - 5)(x + 4)
$$

✔ Answer: $ (2x - 5)(x + 4) $

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✔ 8. $ 2x^2 + 11x - 21 $

$ ac = -42 $, need two numbers that multiply to -42, add to 11 → $ 14 $, $ -3 $

Split:
$$
2x^2 + 14x - 3x - 21 = 2x(x + 7) -3(x + 7) = (2x - 3)(x + 7)
$$

✔ Answer: $ (2x - 3)(x + 7) $

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✔ 9. $ 3y^2 + 13y - 10 $

$ ac = -30 $, need two numbers that multiply to -30, add to 13 → $ 15 $, $ -2 $

Split:
$$
3y^2 + 15y - 2y - 10 = 3y(y + 5) -2(y + 5) = (3y - 2)(y + 5)
$$

✔ Answer: $ (3y - 2)(y + 5) $

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✔ 10. $ 3x^2 + 17x - 20 $

$ ac = -60 $, need two numbers that multiply to -60, add to 17 → $ 20 $, $ -3 $

Split:
$$
3x^2 + 20x - 3x - 20 = x(3x + 20) -1(3x + 20) = (3x - 1)(x + 20)? Wait — let's check:

Wait: $ 3x^2 + 20x - 3x - 20 = 3x^2 + 17x - 20 $

Group:
$$
(3x^2 + 20x) - (3x + 20) → not helpful.

Try factoring directly: Try $ (3x - 4)(x + 5) = 3x^2 + 15x - 4x - 20 = 3x^2 + 11x - 20 $ ✘

Try $ (3x + 5)(x - 4) = 3x^2 -12x + 5x - 20 = 3x^2 -7x -20 $ ✘

Try $ (3x - 5)(x + 4) = 3x^2 + 12x -5x -20 = 3x^2 +7x -20 $ ✘

Try $ (3x + 20)(x - 1) = 3x^2 -3x + 20x -20 = 3x^2 +17x -20 $ ✔

Yes!

✔ Answer: $ (3x + 20)(x - 1) $

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Now let's jump ahead to #27, since many are similar.

But to save time, here’s a summary table of the fully factored forms. I'll solve them efficiently using the AC method.

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## 📘 Full Solutions (Factored Completely)

| Problem | Factored Form |
|--------|----------------|
| 1. $ 2x^2 + 5x + 3 $ | $ (2x + 3)(x + 1) $ |
| 2. $ 2x^2 + 5x + 2 $ | $ (2x + 1)(x + 2) $ |
| 3. $ 2y^2 - 13y + 20 $ | $ (2y - 5)(y - 4) $ |
| 4. $ 2y^2 + 11y + 15 $ | $ (2y + 5)(y + 3) $ |
| 5. $ 2t^2 + 7t - 15 $ | $ (2t - 3)(t + 5) $ |
| 6. $ 2t^2 - 9t - 35 $ | $ (2t + 5)(t - 7) $ |
| 7. $ 2x^2 + 3x - 20 $ | $ (2x - 5)(x + 4) $ |
| 8. $ 2x^2 + 11x - 21 $ | $ (2x - 3)(x + 7) $ |
| 9. $ 3y^2 + 13y - 10 $ | $ (3y - 2)(y + 5) $ |
| 10. $ 3x^2 + 17x - 20 $ | $ (3x + 20)(x - 1) $ |
| 11. $ 3y^2 - 17y - 28 $ | $ (3y + 4)(y - 7) $ |
| 12. $ 3y^2 + 13y + 14 $ | $ (3y + 7)(y + 2) $ |
| 13. $ 5y^2 - 23y + 24 $ | $ (5y - 8)(y - 3) $ |
| 14. $ 5x^2 - 12x - 32 $ | $ (5x + 8)(x - 4) $ |
| 15. $ 5y^2 + 17y + 14 $ | $ (5y + 7)(y + 2) $ |
| 16. $ 5y^2 + 11y - 12 $ | $ (5y - 4)(y + 3) $ |
| 17. $ 4x^2 + 25x + 25 $ | $ (4x + 5)(x + 5) $ |
| 18. $ 4y^2 + 5y - 12 $ | $ (4y - 3)(y + 4) $ |
| 19. $ 4y^2 + 4y - 15 $ | $ (2y - 3)(2y + 5) $ |
| 20. $ 4x^2 - 4x - 35 $ | $ (2x - 7)(2x + 5) $ |
| 21. $ 6x^2 + 7x - 20 $ | $ (3x - 4)(2x + 5) $ |
| 22. $ 6y^2 + 5y - 21 $ | $ (3y - 7)(2y + 3) $ |
| 23. $ 8y^2 + 14y - 15 $ | $ (4y - 3)(2y + 5) $ |
| 24. $ 8x^2 + 6x - 5 $ | $ (4x - 1)(2x + 5) $ |
| 25. $ 12y^2 - y - 6 $ | $ (3y - 2)(4y + 3) $ |
| 26. $ 15y^2 + y - 2 $ | $ (3y - 1)(5y + 2) $ |
| 27. $ 22x^2 - 29x - 6 $ | $ (11x + 2)(2x - 3) $ |
| 28. $ 20z^2 + 7z - 6 $ | $ (5z - 2)(4z + 3) $ |
| 29. $ 2x^2 - 1xy - 10y^2 $ | $ (2x + 5y)(x - 2y) $ |
| 30. $ 2x^2 + 11xy + 12y^2 $ | $ (2x + 3y)(x + 4y) $ |
| 31. $ 3x^2 - 28xy + 32y^2 $ | $ (3x - 4y)(x - 8y) $ |
| 32. $ 3x^2 + 13xy - 10y^2 $ | $ (3x - 2y)(x + 5y) $ |
| 33. $ 5x^2 + 27xy + 10y^2 $ | $ (5x + 2y)(x + 5y) $ |
| 34. $ 5x^2 - 6xy - 8y^2 $ | $ (5x + 4y)(x - 2y) $ |
| 35. $ 7x^2 - 10xy + 3y^2 $ | $ (7x - 3y)(x - y) $ |
| 36. $ 6x^2 + 7xy - 3y^2 $ | $ (3x - y)(2x + 3y) $ |
| 37. $ 2x^3 + 5x^2 - 12x $ | $ x(2x - 3)(x + 4) $ |
| 38. $ 3x^3 - 19x^2 + 20x $ | $ x(3x - 4)(x - 5) $ |
| 39. $ 36x^3 - 12x^2 - 15x $ | $ 3x(6x + 5)(2x - 1) $ |
| 40. $ 6x^3 - 10x^2 - 4x $ | $ 2x(3x + 1)(x - 2) $ |
| 41. $ 18x^3 - 21x^2 - 9x $ | $ 3x(3x + 1)(2x - 3) $ |
| 42. $ 12t^3 - 10t^2 - 12t $ | $ 2t(3t + 2)(2t - 3) $ |
| 43. $ 12t^3 - 22t^2 + 6t $ | $ 2t(3t - 1)(2t - 3) $ |
| 44. $ 15t^3 - 18t^2 - 24t $ | $ 3t(5t + 4)(t - 2) $ |
| 45. $ 5x^3y - 10x^2y^2 - 15xy^3 $ | $ 5xy(x - 2y - 3y^2) $? Wait — factor out common terms first |

Let’s do #45 carefully:

✔ 45. $ 5x^3y - 10x^2y^2 - 15xy^3 $

Factor out GCF: $ 5xy $

$$
= 5xy(x^2 - 2xy - 3y^2)
$$

Now factor $ x^2 - 2xy - 3y^2 $: two numbers that multiply to -3, add to -2 → $ -3 $, $ 1 $

So:
$$
x^2 - 2xy - 3y^2 = (x - 3y)(x + y)
$$

✔ Final: $ 5xy(x - 3y)(x + y) $

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✔ 46. $ 6x^5y + 25x^4y^2 + 4x^3y^3 $

GCF: $ x^3y $

$$
= x^3y(6x^2 + 25xy + 4y^2)
$$

Now factor $ 6x^2 + 25xy + 4y^2 $: $ ac = 24 $, need two numbers that multiply to 24, add to 25 → $ 24 $, $ 1 $

So:
$$
6x^2 + 24xy + xy + 4y^2 = 6x(x + 4y) + y(x + 4y) = (6x + y)(x + 4y)
$$

✔ Final: $ x^3y(6x + y)(x + 4y) $

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✔ 47. $ 12x^4y^3 + 11x^3y^4 + 2x^2y^5 $

GCF: $ x^2y^3 $

$$
= x^2y^3(12x^2 + 11xy + 2y^2)
$$

Factor $ 12x^2 + 11xy + 2y^2 $: $ ac = 24 $, need two numbers that multiply to 24, add to 11 → $ 8 $, $ 3 $

$$
12x^2 + 8xy + 3xy + 2y^2 = 4x(3x + 2y) + y(3x + 2y) = (4x + y)(3x + 2y)
$$

✔ Final: $ x^2y^3(4x + y)(3x + 2y) $

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✔ 48. $ 12x^3y^3 + 28x^2y^4 + 8xy^5 $

GCF: $ 4xy^3 $

$$
= 4xy^3(3x^2 + 7xy + 2y^2)
$$

Factor $ 3x^2 + 7xy + 2y^2 $: $ ac = 6 $, need two numbers that multiply to 6, add to 7 → $ 6 $, $ 1 $

$$
3x^2 + 6xy + xy + 2y^2 = 3x(x + 2y) + y(x + 2y) = (3x + y)(x + 2y)
$$

✔ Final: $ 4xy^3(3x + y)(x + 2y) $

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✔ 49. $ -x^3 - 5x^2 - 6x $

Factor out $ -x $:

$$
= -x(x^2 + 5x + 6) = -x(x + 2)(x + 3)
$$

✔ Answer: $ -x(x + 2)(x + 3) $

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✔ 50. $ -y^3 + 3y^2 - 2y $

Factor out $ -y $:

$$
= -y(y^2 - 3y + 2) = -y(y - 1)(y - 2)
$$

✔ Answer: $ -y(y - 1)(y - 2) $

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✔ 51. $ -6x^2 - 5x + 6 $

Factor out $ -1 $: $ -(6x^2 + 5x - 6) $

Now factor $ 6x^2 + 5x - 6 $: $ ac = -36 $, need two numbers that multiply to -36, add to 5 → $ 9 $, $ -4 $

$$
6x^2 + 9x - 4x - 6 = 3x(2x + 3) -2(2x + 3) = (3x - 2)(2x + 3)
$$

So original: $ -(3x - 2)(2x + 3) $

✔ Answer: $ -(3x - 2)(2x + 3) $

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✔ 52. $ -8m^2 + 10mn + 3n^2 $

Factor out $ -1 $: $ -(8m^2 - 10mn - 3n^2) $

Now factor $ 8m^2 - 10mn - 3n^2 $: $ ac = -24 $, need two numbers that multiply to -24, add to -10 → $ -12 $, $ 2 $

$$
8m^2 -12mn + 2mn - 3n^2 = 4m(2m - 3n) + n(2m - 3n) = (4m + n)(2m - 3n)
$$

So: $ -(4m + n)(2m - 3n) $

✔ Answer: $ -(4m + n)(2m - 3n) $

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## ✔ Final Answer Summary (All Problems Solved)

Here are the fully factored answers:

1. $ (2x + 3)(x + 1) $
2. $ (2x + 1)(x + 2) $
3. $ (2y - 5)(y - 4) $
4. $ (2y + 5)(y + 3) $
5. $ (2t - 3)(t + 5) $
6. $ (2t + 5)(t - 7) $
7. $ (2x - 5)(x + 4) $
8. $ (2x - 3)(x + 7) $
9. $ (3y - 2)(y + 5) $
10. $ (3x + 20)(x - 1) $
11. $ (3y + 4)(y - 7) $
12. $ (3y + 7)(y + 2) $
13. $ (5y - 8)(y - 3) $
14. $ (5x + 8)(x - 4) $
15. $ (5y + 7)(y + 2) $
16. $ (5y - 4)(y + 3) $
17. $ (4x + 5)(x + 5) $
18. $ (4y - 3)(y + 4) $
19. $ (2y - 3)(2y + 5) $
20. $ (2x - 7)(2x + 5) $
21. $ (3x - 4)(2x + 5) $
22. $ (3y - 7)(2y + 3) $
23. $ (4y - 3)(2y + 5) $
24. $ (4x - 1)(2x + 5) $
25. $ (3y - 2)(4y + 3) $
26. $ (3y - 1)(5y + 2) $
27. $ (11x + 2)(2x - 3) $
28. $ (5z - 2)(4z + 3) $
29. $ (2x + 5y)(x - 2y) $
30. $ (2x + 3y)(x + 4y) $
31. $ (3x - 4y)(x - 8y) $
32. $ (3x - 2y)(x + 5y) $
33. $ (5x + 2y)(x + 5y) $
34. $ (5x + 4y)(x - 2y) $
35. $ (7x - 3y)(x - y) $
36. $ (3x - y)(2x + 3y) $
37. $ x(2x - 3)(x + 4) $
38. $ x(3x - 4)(x - 5) $
39. $ 3x(6x + 5)(2x - 1) $
40. $ 2x(3x + 1)(x - 2) $
41. $ 3x(3x + 1)(2x - 3) $
42. $ 2t(3t + 2)(2t - 3) $
43. $ 2t(3t - 1)(2t - 3) $
44. $ 3t(5t + 4)(t - 2) $
45. $ 5xy(x - 3y)(x + y) $
46. $ x^3y(6x + y)(x + 4y) $
47. $ x^2y^3(4x + y)(3x + 2y) $
48. $ 4xy^3(3x + y)(x + 2y) $
49. $ -x(x + 2)(x + 3) $
50. $ -y(y - 1)(y - 2) $
51. $ -(3x - 2)(2x + 3) $
52. $ -(4m + n)(2m - 3n) $

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Let me know if you'd like step-by-step explanations for any specific problem!