10 Printable Math Worksheets - Free Printable

Let's solve this 9th Grade Math Worksheet step by step. The goal is to factor each quadratic expression.

We'll go through Section A, Section B, and Section C, using standard factoring techniques:

---

🔧 Factoring Techniques Used:

1. For quadratics in the form $ ax^2 + bx + c $:
- If $ a = 1 $: Find two numbers that multiply to $ c $ and add to $ b $.
- If $ a \neq 1 $: Use splitting the middle term (also called the "AC method") or trial and error.

---

## ✔ SECTION A

Factor the following:

---

1) $ 2x^2 + x - 1 $

We need two numbers that multiply to $ 2 \times (-1) = -2 $ and add to $ 1 $.
Try: $ 2 $ and $ -1 $ → $ 2 \times (-1) = -2 $, $ 2 + (-1) = 1 $ → works!

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

✔ Answer: $ (2x - 1)(x + 1) $

---

2) $ 2x^2 + x - 3 $

Multiply $ 2 \times (-3) = -6 $. Need two numbers that multiply to $-6$ and add to $1$: $ 3 $ and $-2$

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

✔ Answer: $ (x - 1)(2x + 3) $

---

3) $ 2x^2 + 9x - 5 $

$ a = 2, c = -5 $, so $ ac = -10 $. Need numbers that multiply to $-10$, add to $9$: $10$ and $-1$

Split:
$$
2x^2 + 10x - x - 5 = 2x(x + 5) -1(x + 5) = (2x - 1)(x + 5)
$$

✔ Answer: $ (2x - 1)(x + 5) $

---

4) $ 2x^2 - 3x - 2 $

$ ac = 2 \times (-2) = -4 $. Need numbers: $-4$ and $1$ → sum = $-3$

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

✔ Answer: $ (2x + 1)(x - 2) $

---

5) $ 2x^2 - 13x - 24 $

$ ac = 2 \times (-24) = -48 $. Need two numbers: $-16$ and $3$ → $-16 \times 3 = -48$, $-16 + 3 = -13$

Split:
$$
2x^2 - 16x + 3x - 24 = 2x(x - 8) +3(x - 8) = (2x + 3)(x - 8)
$$

✔ Answer: $ (2x + 3)(x - 8) $

---

6) $ 3x^2 - 14x - 5 $

$ ac = 3 \times (-5) = -15 $. Need: $-15$ and $1$ → sum = $-14$

Split:
$$
3x^2 - 15x + x - 5 = 3x(x - 5) +1(x - 5) = (3x + 1)(x - 5)
$$

✔ Answer: $ (3x + 1)(x - 5) $

---

7) $ 3x^2 - 8x - 11 $

$ ac = 3 \times (-11) = -33 $. Try $-11$ and $3$: $-11 + 3 = -8$

Split:
$$
3x^2 - 11x + 3x - 11 = x(3x - 11) +1(3x - 11) = (x + 1)(3x - 11)
$$

✔ Answer: $ (x + 1)(3x - 11) $

---

8) $ 2x^2 - 14x + 12 $

First, factor out GCF: $ 2(x^2 - 7x + 6) $

Now factor $ x^2 - 7x + 6 $: $-6$ and $-1$: $-6 \times -1 = 6$, $-6 + (-1) = -7$

→ $ (x - 6)(x - 1) $

So total: $ 2(x - 6)(x - 1) $

✔ Answer: $ 2(x - 6)(x - 1) $

---

9) $ 3x^2 - 21x + 36 $

GCF: 3 → $ 3(x^2 - 7x + 12) $

Factor $ x^2 - 7x + 12 $: $-3$ and $-4$: $-3 \times -4 = 12$, $-3 + (-4) = -7$

→ $ (x - 3)(x - 4) $

So: $ 3(x - 3)(x - 4) $

✔ Answer: $ 3(x - 3)(x - 4) $

---

10) $ 5x^2 - 41x + 8 $

$ ac = 5 \times 8 = 40 $. Need two numbers that multiply to $40$, add to $-41$: $-40$ and $-1$

Split:
$$
5x^2 - 40x - x + 8 = 5x(x - 8) -1(x - 8) = (5x - 1)(x - 8)
$$

✔ Answer: $ (5x - 1)(x - 8) $

---

11) $ 3x^2 - 2x - 21 $

$ ac = 3 \times (-21) = -63 $. Try $-9$ and $7$: $-9 \times 7 = -63$, $-9 + 7 = -2$

Split:
$$
3x^2 - 9x + 7x - 21 = 3x(x - 3) +7(x - 3) = (3x + 7)(x - 3)
$$

✔ Answer: $ (3x + 7)(x - 3) $

---

12) $ 2x^2 + 2x - 12 $

GCF: 2 → $ 2(x^2 + x - 6) $

Factor $ x^2 + x - 6 $: $3$ and $-2$: $3 \times -2 = -6$, $3 + (-2) = 1$

→ $ (x + 3)(x - 2) $

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

✔ Answer: $ 2(x + 3)(x - 2) $

---

13) $ 2x^2 - 11x + 15 $

$ ac = 2 \times 15 = 30 $. Need: $-6$ and $-5$: $-6 \times -5 = 30$, $-6 + (-5) = -11$

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

✔ Answer: $ (2x - 5)(x - 3) $

---

14) $ 3x^2 - 34x - 24 $

$ ac = 3 \times (-24) = -72 $. Try $-36$ and $2$: $-36 \times 2 = -72$, $-36 + 2 = -34$

Split:
$$
3x^2 - 36x + 2x - 24 = 3x(x - 12) +2(x - 12) = (3x + 2)(x - 12)
$$

✔ Answer: $ (3x + 2)(x - 12) $

---

15) $ 5x^2 - 27x + 10 $

$ ac = 5 \times 10 = 50 $. Need: $-25$ and $-2$: $-25 \times -2 = 50$, $-25 + (-2) = -27$

Split:
$$
5x^2 - 25x - 2x + 10 = 5x(x - 5) -2(x - 5) = (5x - 2)(x - 5)
$$

✔ Answer: $ (5x - 2)(x - 5) $

---

## ✔ SECTION B

All have $ a = 1 $, so easier — just find two numbers that multiply to $ c $, add to $ b $.

---

1) $ x^2 + 7x - 30 $

Need: $10$ and $-3$: $10 \times -3 = -30$, $10 + (-3) = 7$

→ $ (x + 10)(x - 3) $

✔ Answer: $ (x + 10)(x - 3) $

---

2) $ x^2 + 9x + 20 $

$4$ and $5$: $4 \times 5 = 20$, $4 + 5 = 9$

→ $ (x + 4)(x + 5) $

✔ Answer: $ (x + 4)(x + 5) $

---

3) $ x^2 + 8x - 9 $

$9$ and $-1$: $9 \times -1 = -9$, $9 + (-1) = 8$

→ $ (x + 9)(x - 1) $

✔ Answer: $ (x + 9)(x - 1) $

---

4) $ x^2 - 18x + 80 $

$-10$ and $-8$: $-10 \times -8 = 80$, $-10 + (-8) = -18$

→ $ (x - 10)(x - 8) $

✔ Answer: $ (x - 10)(x - 8) $

---

5) $ x^2 - 11x + 28 $

$-7$ and $-4$: $-7 \times -4 = 28$, $-7 + (-4) = -11$

→ $ (x - 7)(x - 4) $

✔ Answer: $ (x - 7)(x - 4) $

---

6) $ x^2 + 6x - 72 $

Need: $12$ and $-6$: $12 \times -6 = -72$, $12 + (-6) = 6$

→ $ (x + 12)(x - 6) $

✔ Answer: $ (x + 12)(x - 6) $

---

7) $ x^2 - 9x - 22 $

$-11$ and $2$: $-11 \times 2 = -22$, $-11 + 2 = -9$

→ $ (x - 11)(x + 2) $

✔ Answer: $ (x - 11)(x + 2) $

---

8) $ x^2 - x - 12 $

$-4$ and $3$: $-4 \times 3 = -12$, $-4 + 3 = -1$

→ $ (x - 4)(x + 3) $

✔ Answer: $ (x - 4)(x + 3) $

---

9) $ x^2 + 3x - 108 $

Find two numbers: $12$ and $-9$: $12 \times -9 = -108$, $12 + (-9) = 3$

→ $ (x + 12)(x - 9) $

✔ Answer: $ (x + 12)(x - 9) $

---

10) $ x^2 - 17x + 72 $

$-8$ and $-9$: $-8 \times -9 = 72$, $-8 + (-9) = -17$

→ $ (x - 8)(x - 9) $

✔ Answer: $ (x - 8)(x - 9) $

---

11) $ x^2 - x - 42 $

$-7$ and $6$: $-7 \times 6 = -42$, $-7 + 6 = -1$

→ $ (x - 7)(x + 6) $

✔ Answer: $ (x - 7)(x + 6) $

---

12) $ x^2 - 15x + 56 $

$-8$ and $-7$: $-8 \times -7 = 56$, $-8 + (-7) = -15$

→ $ (x - 8)(x - 7) $

✔ Answer: $ (x - 8)(x - 7) $

---

## ✔ SECTION C

All have $ a = 2 $, so use splitting method.

---

1) $ 2x^2 + 3x + 1 $

$ ac = 2 \times 1 = 2 $. Numbers: $2$ and $1$ → sum = $3$

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

✔ Answer: $ (2x + 1)(x + 1) $

---

2) $ 2x^2 + 5x + 2 $

$ ac = 4 $. Numbers: $4$ and $1$: $4 + 1 = 5$

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

✔ Answer: $ (2x + 1)(x + 2) $

---

3) $ 2x^2 + 7x + 3 $

$ ac = 6 $. Numbers: $6$ and $1$: $6 + 1 = 7$

Split:
$$
2x^2 + 6x + x + 3 = 2x(x + 3) +1(x + 3) = (2x + 1)(x + 3)
$$

✔ Answer: $ (2x + 1)(x + 3) $

---

4) $ 2x^2 + 7x + 5 $

$ ac = 10 $. Numbers: $5$ and $2$: $5 + 2 = 7$

Split:
$$
2x^2 + 5x + 2x + 5 = x(2x + 5) +1(2x + 5) = (x + 1)(2x + 5)
$$

✔ Answer: $ (x + 1)(2x + 5) $

---

5) $ 2x^2 + 9x + 7 $

$ ac = 14 $. Numbers: $7$ and $2$: $7 + 2 = 9$

Split:
$$
2x^2 + 7x + 2x + 7 = x(2x + 7) +1(2x + 7) = (x + 1)(2x + 7)
$$

✔ Answer: $ (x + 1)(2x + 7) $

---

6) $ 2x^2 + 5x + 3 $

$ ac = 6 $. Numbers: $3$ and $2$: $3 + 2 = 5$

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

✔ Answer: $ (x + 1)(2x + 3) $

---

7) $ 2x^2 + 8x + 6 $

GCF: 2 → $ 2(x^2 + 4x + 3) $

Factor: $ (x + 1)(x + 3) $

So: $ 2(x + 1)(x + 3) $

✔ Answer: $ 2(x + 1)(x + 3) $

---

8) $ 2x^2 + 9x + 10 $

$ ac = 20 $. Numbers: $5$ and $4$: $5 + 4 = 9$

Split:
$$
2x^2 + 5x + 4x + 10 = x(2x + 5) +2(2x + 5) = (x + 2)(2x + 5)
$$

✔ Answer: $ (x + 2)(2x + 5) $

---

9) $ 2x^2 + 16x + 14 $

GCF: 2 → $ 2(x^2 + 8x + 7) $

Factor: $ (x + 1)(x + 7) $

So: $ 2(x + 1)(x + 7) $

✔ Answer: $ 2(x + 1)(x + 7) $

---

10) $ 2x^2 + 16x + 24 $

GCF: 2 → $ 2(x^2 + 8x + 12) $

Factor: $ (x + 6)(x + 2) $

So: $ 2(x + 6)(x + 2) $

✔ Answer: $ 2(x + 6)(x + 2) $

---

11) $ 2x^2 + 12x + 18 $

GCF: 2 → $ 2(x^2 + 6x + 9) = 2(x + 3)^2 $

✔ Answer: $ 2(x + 3)^2 $

---

12) $ 2x^2 + 14x + 20 $

GCF: 2 → $ 2(x^2 + 7x + 10) = 2(x + 2)(x + 5) $

✔ Answer: $ 2(x + 2)(x + 5) $

---

13) $ 2x^2 + 22x + 36 $

GCF: 2 → $ 2(x^2 + 11x + 18) = 2(x + 2)(x + 9) $

✔ Answer: $ 2(x + 2)(x + 9) $

---

14) $ 2x^2 + 28x + 48 $

GCF: 2 → $ 2(x^2 + 14x + 24) = 2(x + 2)(x + 12) $

✔ Answer: $ 2(x + 2)(x + 12) $

---

15) $ 2x^2 + 26x + 72 $

GCF: 2 → $ 2(x^2 + 13x + 36) = 2(x + 4)(x + 9) $

✔ Answer: $ 2(x + 4)(x + 9) $

---

## 📌 Final Answers Summary

SECTION A:

1. $ (2x - 1)(x + 1) $
2. $ (x - 1)(2x + 3) $
3. $ (2x - 1)(x + 5) $
4. $ (2x + 1)(x - 2) $
5. $ (2x + 3)(x - 8) $
6. $ (3x + 1)(x - 5) $
7. $ (x + 1)(3x - 11) $
8. $ 2(x - 6)(x - 1) $
9. $ 3(x - 3)(x - 4) $
10. $ (5x - 1)(x - 8) $
11. $ (3x + 7)(x - 3) $
12. $ 2(x + 3)(x - 2) $
13. $ (2x - 5)(x - 3) $
14. $ (3x + 2)(x - 12) $
15. $ (5x - 2)(x - 5) $

---

SECTION B:

1. $ (x + 10)(x - 3) $
2. $ (x + 4)(x + 5) $
3. $ (x + 9)(x - 1) $
4. $ (x - 10)(x - 8) $
5. $ (x - 7)(x - 4) $
6. $ (x + 12)(x - 6) $
7. $ (x - 11)(x + 2) $
8. $ (x - 4)(x + 3) $
9. $ (x + 12)(x - 9) $
10. $ (x - 8)(x - 9) $
11. $ (x - 7)(x + 6) $
12. $ (x - 8)(x - 7) $

---

SECTION C:

1. $ (2x + 1)(x + 1) $
2. $ (2x + 1)(x + 2) $
3. $ (2x + 1)(x + 3) $
4. $ (x + 1)(2x + 5) $
5. $ (x + 1)(2x + 7) $
6. $ (x + 1)(2x + 3) $
7. $ 2(x + 1)(x + 3) $
8. $ (x + 2)(2x + 5) $
9. $ 2(x + 1)(x + 7) $
10. $ 2(x + 6)(x + 2) $
11. $ 2(x + 3)^2 $
12. $ 2(x + 2)(x + 5) $
13. $ 2(x + 2)(x + 9) $
14. $ 2(x + 2)(x + 12) $
15. $ 2(x + 4)(x + 9) $

---

Let me know if you'd like these checked with multiplication or explained further!