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 $.

Below is a step-by-step explanation and solution for each problem. I'll go through several examples to demonstrate the method, then provide the answers for all 52 problems.

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🔧 Method: Factoring Trinomials of the Form $ ax^2 + bx + c $

We use the "AC Method":

1. Multiply $ a \times c $
2. Find two numbers that:
- Multiply to $ a \cdot c $
- Add to $ b $
3. Split the middle term using those two numbers
4. Factor by grouping

Alternatively, for simpler cases where $ a = 1 $, we can factor directly.

Let’s solve a few step-by-step first.

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✔ Example 1: $ 2x^2 + 5x + 3 $

- $ a = 2, b = 5, c = 3 $
- $ ac = 2 \times 3 = 6 $
- Find two numbers that multiply to 6 and add to 5 → $ 2 $ and $ 3 $
- Rewrite:
$ 2x^2 + 2x + 3x + 3 $
- Group:
$ (2x^2 + 2x) + (3x + 3) = 2x(x + 1) + 3(x + 1) $
- Factor:
$ (2x + 3)(x + 1) $

✔ Answer: $ (2x + 3)(x + 1) $

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✔ Example 2: $ 2x^2 + 5x + 2 $

- $ a = 2, b = 5, c = 2 $
- $ ac = 4 $
- Numbers: $ 4 $ and $ 1 $ (since $ 4 \times 1 = 4 $, $ 4 + 1 = 5 $)
- Rewrite: $ 2x^2 + 4x + x + 2 $
- Group: $ (2x^2 + 4x) + (x + 2) = 2x(x + 2) + 1(x + 2) $
- Factor: $ (2x + 1)(x + 2) $

✔ Answer: $ (2x + 1)(x + 2) $

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✔ Example 3: $ 2y^2 - 13y + 20 $

- $ a = 2, b = -13, c = 20 $
- $ ac = 40 $
- Need two numbers that multiply to 40 and add to -13 → $ -8 $ and $ -5 $
- Rewrite: $ 2y^2 - 8y - 5y + 20 $
- Group: $ (2y^2 - 8y) + (-5y + 20) = 2y(y - 4) -5(y - 4) $
- Factor: $ (2y - 5)(y - 4) $

✔ Answer: $ (2y - 5)(y - 4) $

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Now let's go through all 52 problems, factoring completely.

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📝 Full Solutions

#### 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 $
$ ac = 30 $, numbers: 6 and 5 → $ (2y + 5)(y + 3) $

#### 5. $ 2t^2 + 7t - 15 $
$ ac = -30 $, numbers: 10 and -3 → $ (2t - 3)(t + 5) $

#### 6. $ 2t^2 - 9t - 35 $
$ ac = -70 $, numbers: -14 and 5 → $ (2t + 5)(t - 7) $

#### 7. $ 2x^2 + 3x - 20 $
$ ac = -40 $, numbers: 8 and -5 → $ (2x - 5)(x + 4) $

#### 8. $ 2x^2 + 11x - 21 $
$ ac = -42 $, numbers: 14 and -3 → $ (2x - 3)(x + 7) $

#### 9. $ 3y^2 + 13y - 10 $
$ ac = -30 $, numbers: 15 and -2 → $ (3y - 2)(y + 5) $

#### 10. $ 3x^2 + 17x - 20 $
$ ac = -60 $, numbers: 20 and -3 → $ (3x - 4)(x + 5) $

#### 11. $ 3y^2 - 17y - 28 $
$ ac = -84 $, numbers: -21 and 4 → $ (3y + 4)(y - 7) $

#### 12. $ 3y^2 + 13y + 14 $
$ ac = 42 $, numbers: 7 and 6 → $ (3y + 7)(y + 2) $

#### 13. $ 5y^2 - 23y + 24 $
$ ac = 120 $, numbers: -15 and -8 → $ (5y - 8)(y - 3) $

#### 14. $ 5x^2 - 12x - 32 $
$ ac = -160 $, numbers: -20 and 8 → $ (5x + 8)(x - 4) $

#### 15. $ 5y^2 + 17y + 14 $
$ ac = 70 $, numbers: 10 and 7 → $ (5y + 7)(y + 2) $

#### 16. $ 5y^2 + 11y - 12 $
$ ac = -60 $, numbers: 15 and -4 → $ (5y - 4)(y + 3) $

#### 17. $ 4x^2 + 25x + 25 $
$ ac = 100 $, numbers: 20 and 5 → $ (4x + 5)(x + 5) $

#### 18. $ 4y^2 + 5y - 12 $
$ ac = -48 $, numbers: 9 and -4 → $ (4y - 3)(y + 4) $

#### 19. $ 4y^2 + 4y - 15 $
$ ac = -60 $, numbers: 10 and -6 → $ (2y - 3)(2y + 5) $

#### 20. $ 4x^2 - 4x - 35 $
$ ac = -140 $, numbers: 10 and -14 → $ (2x + 5)(2x - 7) $

#### 21. $ 6x^2 + 7x - 20 $
$ ac = -120 $, numbers: 15 and -8 → $ (3x - 4)(2x + 5) $

#### 22. $ 6y^2 + 5y - 21 $
$ ac = -126 $, numbers: 14 and -9 → $ (3y - 7)(2y + 3) $

#### 23. $ 8y^2 + 14y - 15 $
$ ac = -120 $, numbers: 20 and -6 → $ (4y - 3)(2y + 5) $

#### 24. $ 8x^2 + 6x - 5 $
$ ac = -40 $, numbers: 10 and -4 → $ (4x - 1)(2x + 5) $

#### 25. $ 12y^2 - y - 6 $
$ ac = -72 $, numbers: 8 and -9 → $ (3y - 2)(4y + 3) $

#### 26. $ 15y^2 + y - 2 $
$ ac = -30 $, numbers: 6 and -5 → $ (5y - 2)(3y + 1) $

#### 27. $ 22x^2 - 29x - 6 $
$ ac = -132 $, numbers: -33 and 4 → $ (11x + 2)(2x - 3) $

#### 28. $ 20z^2 + 7z - 6 $
$ ac = -120 $, numbers: 15 and -8 → $ (5z - 2)(4z + 3) $

#### 29. $ 2x^2 - 1xy - 10y^2 $
Treat as $ 2x^2 - xy - 10y^2 $, $ ac = -20 $, numbers: 4 and -5 → $ (2x + 4y)(x - 5y) $? Wait — check:

Better: $ 2x^2 - xy - 10y^2 $

Try: $ (2x + 5y)(x - 2y) = 2x^2 - 4xy + 5xy - 10y^2 = 2x^2 + xy - 10y^2 $ ✘

Try: $ (2x - 5y)(x + 2y) = 2x^2 + 4xy - 5xy - 10y^2 = 2x^2 - xy - 10y^2 $ ✔

✔ Answer: $ (2x - 5y)(x + 2y) $

#### 30. $ 2x^2 + 11xy + 12y^2 $
$ ac = 24 $, numbers: 8 and 3 → $ (2x + 3y)(x + 4y) $

#### 31. $ 3x^2 - 28xy + 32y^2 $
$ ac = 96 $, numbers: -24 and -4 → $ (3x - 4y)(x - 8y) $

#### 32. $ 3x^2 + 13xy - 10y^2 $
$ ac = -30 $, numbers: 15 and -2 → $ (3x - 2y)(x + 5y) $

#### 33. $ 5x^2 + 27xy + 10y^2 $
$ ac = 50 $, numbers: 25 and 2 → $ (5x + 2y)(x + 5y) $

#### 34. $ 5x^2 - 6xy - 8y^2 $
$ ac = -40 $, numbers: 4 and -10 → $ (5x + 4y)(x - 2y) $

#### 35. $ 7x^2 - 10xy + 3y^2 $
$ ac = 21 $, numbers: -7 and -3 → $ (7x - 3y)(x - y) $

#### 36. $ 6x^2 + 7xy - 3y^2 $
$ ac = -18 $, numbers: 9 and -2 → $ (3x - y)(2x + 3y) $

#### 37. $ 2x^3 + 5x^2 - 12x $
Factor out $ x $: $ x(2x^2 + 5x - 12) $
Then: $ 2x^2 + 5x - 12 $ → $ ac = -24 $, numbers: 8 and -3 → $ (2x - 3)(x + 4) $
✔ $ x(2x - 3)(x + 4) $

#### 38. $ 3x^3 - 19x^2 + 20x $
Factor $ x $: $ x(3x^2 - 19x + 20) $
$ ac = 60 $, numbers: -15 and -4 → $ (3x - 4)(x - 5) $
✔ $ x(3x - 4)(x - 5) $

#### 39. $ 36x^3 - 12x^2 - 15x $
Factor $ 3x $: $ 3x(12x^2 - 4x - 5) $
$ ac = -60 $, numbers: 6 and -10 → $ (6x + 5)(2x - 1) $? Check:

Wait: $ 12x^2 - 4x - 5 $

Try: $ (6x + 5)(2x - 1) = 12x^2 - 6x + 10x - 5 = 12x^2 + 4x - 5 $ ✘

Try: $ (6x - 5)(2x + 1) = 12x^2 + 6x - 10x - 5 = 12x^2 - 4x - 5 $ ✔

So: $ 3x(6x - 5)(2x + 1) $

✔ $ 3x(6x - 5)(2x + 1) $

#### 40. $ 6x^3 - 10x^2 - 4x $
Factor $ 2x $: $ 2x(3x^2 - 5x - 2) $
$ ac = -6 $, numbers: -6 and 1 → $ (3x + 1)(x - 2) $
✔ $ 2x(3x + 1)(x - 2) $

#### 41. $ 18x^3 - 21x^2 - 9x $
Factor $ 3x $: $ 3x(6x^2 - 7x - 3) $
$ ac = -18 $, numbers: -9 and 2 → $ (3x + 1)(2x - 3) $
✔ $ 3x(3x + 1)(2x - 3) $

#### 42. $ 12t^3 - 10t^2 - 12t $
Factor $ 2t $: $ 2t(6t^2 - 5t - 6) $
$ ac = -36 $, numbers: -9 and 4 → $ (3t + 2)(2t - 3) $
✔ $ 2t(3t + 2)(2t - 3) $

#### 43. $ 12t^3 - 22t^2 + 6t $
Factor $ 2t $: $ 2t(6t^2 - 11t + 3) $
$ ac = 18 $, numbers: -9 and -2 → $ (3t - 1)(2t - 3) $
✔ $ 2t(3t - 1)(2t - 3) $

#### 44. $ 15t^3 - 18t^2 - 24t $
Factor $ 3t $: $ 3t(5t^2 - 6t - 8) $
$ ac = -40 $, numbers: -10 and 4 → $ (5t + 4)(t - 2) $
✔ $ 3t(5t + 4)(t - 2) $

#### 45. $ 5x^3y - 10x^2y^2 - 15xy^3 $
Factor $ 5xy $: $ 5xy(x^2 - 2xy - 3y^2) $
Now: $ x^2 - 2xy - 3y^2 = (x - 3y)(x + y) $
✔ $ 5xy(x - 3y)(x + y) $

#### 46. $ 6x^2y + 25x^4y^2 + 4x^3y^3 $
Rearrange: $ 25x^4y^2 + 4x^3y^3 + 6x^2y $
Factor $ x^2y $: $ x^2y(25x^2 + 4xy + 6) $
Check if quadratic factors: $ 25x^2 + 4xy + 6 $
Discriminant: $ 16y^2 - 600 $ → not perfect square → does not factor further over integers

✔ $ x^2y(25x^2 + 4xy + 6) $

#### 47. $ 12x^4y^3 + 11x^3y^4 + 2x^2y^5 $
Factor $ x^2y^3 $: $ x^2y^3(12x^2 + 11xy + 2y^2) $
Now: $ 12x^2 + 11xy + 2y^2 $ → $ ac = 24 $, numbers: 8 and 3 → $ (3x + y)(4x + 2y) $? Try:

$ (3x + y)(4x + 2y) = 12x^2 + 6xy + 4xy + 2y^2 = 12x^2 + 10xy + 2y^2 $ ✘

Try: $ (3x + 2y)(4x + y) = 12x^2 + 3xy + 8xy + 2y^2 = 12x^2 + 11xy + 2y^2 $ ✔

✔ $ x^2y^3(3x + 2y)(4x + y) $

#### 48. $ 12x^3y^3 + 28x^2y^4 + 8xy^5 $
Factor $ 4xy^3 $: $ 4xy^3(3x^2 + 7xy + 2y^2) $
Now: $ 3x^2 + 7xy + 2y^2 $ → $ ac = 6 $, numbers: 6 and 1 → $ (3x + y)(x + 2y) $
✔ $ 4xy^3(3x + y)(x + 2y) $

#### 49. $ -x^3 - 5x^2 - 6x $
Factor $ -x $: $ -x(x^2 + 5x + 6) = -x(x + 2)(x + 3) $

✔ $ -x(x + 2)(x + 3) $

#### 50. $ -y^3 + 3y^2 - 2y $
Factor $ -y $: $ -y(y^2 - 3y + 2) = -y(y - 1)(y - 2) $

✔ $ -y(y - 1)(y - 2) $

#### 51. $ -6x^2 - 5x + 6 $
Factor out $ -1 $: $ -(6x^2 + 5x - 6) $
Now: $ 6x^2 + 5x - 6 $, $ ac = -36 $, numbers: 9 and -4 → $ (3x - 2)(2x + 3) $
✔ $ -(3x - 2)(2x + 3) $

#### 52. $ -8m^2 + 10mn + 3n^2 $
Factor $ -1 $: $ -(8m^2 - 10mn - 3n^2) $
Now: $ 8m^2 - 10mn - 3n^2 $, $ ac = -24 $, numbers: -12 and 2 → $ (4m + n)(2m - 3n) $? Check:

Try: $ (4m + n)(2m - 3n) = 8m^2 - 12mn + 2mn - 3n^2 = 8m^2 - 10mn - 3n^2 $ ✔

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

✔ $ -(4m + n)(2m - 3n) $

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✔ Final Answers Summary

| Problem | Factored Form |
|--------|----------------|
| 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 - 4)(x + 5) $ |
| 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 + 5)(2x - 7) $ |
| 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 | $ (5y - 2)(3y + 1) $ |
| 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^2y(25x^2 + 4xy + 6) $ |
| 47 | $ x^2y^3(3x + 2y)(4x + y) $ |
| 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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If you need any specific problem explained in more detail, just ask!