Factoring Quadratics Worksheets (printable, online, answers, examples) - Free Printable

Let’s factor each quadratic expression completely. We’re looking for two binomials that multiply to give the original trinomial. Since all coefficients are positive and a > 1, we’ll use the “ac method” or trial-and-error with factors.

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1. Factor: 5n² + 19n + 12

We need two numbers that:
- Multiply to (5 × 12) = 60
- Add to 19

Factors of 60:
(1,60), (2,30), (3,20), (4,15), (5,12), (6,10)

→ 4 and 15 work! (4 + 15 = 19)

Now split the middle term:

5n² + 4n + 15n + 12

Group:

(5n² + 4n) + (15n + 12)

Factor each group:

n(5n + 4) + 3(5n + 4)

Common factor: (5n + 4)

→ (n + 3)(5n + 4)

✔ Check: n×5n=5n², n×4=4n, 3×5n=15n, 3×4=12 → 5n²+19n+12 ✔️

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2. Factor: 9k² + 66k + 21

First, check for GCF: All divisible by 3.

→ 3(3k² + 22k + 7)

Now factor inside: 3k² + 22k + 7

Need two numbers that:
- Multiply to (3 × 7) = 21
- Add to 22

→ 21 and 1 work!

Split middle term:

3k² + 21k + k + 7

Group:

(3k² + 21k) + (k + 7)

Factor:

3k(k + 7) + 1(k + 7)

→ (3k + 1)(k + 7)

Don’t forget the 3 we factored out earlier!

Final: 3(3k + 1)(k + 7)

✔ Check: 3 × [3k×k=3k², 3k×7=21k, 1×k=k, 1×7=7] → 3(3k²+22k+7)=9k²+66k+21 ✔️

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3. Factor: 2n² + 5n + 2

Multiply a×c = 2×2 = 4
Find two numbers that multiply to 4 and add to 5 → 4 and 1

Split:

2n² + 4n + n + 2

Group:

(2n² + 4n) + (n + 2)

Factor:

2n(n + 2) + 1(n + 2)

→ (2n + 1)(n + 2)

✔ Check: 2n×n=2n², 2n×2=4n, 1×n=n, 1×2=2 → 2n²+5n+2 ✔️

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4. Factor: 6x² + 37x + 6

a×c = 6×6 = 36
Need two numbers that multiply to 36 and add to 37 → 36 and 1

Split:

6x² + 36x + x + 6

Group:

(6x² + 36x) + (x + 6)

Factor:

6x(x + 6) + 1(x + 6)

→ (6x + 1)(x + 6)

✔ Check: 6x×x=6x², 6x×6=36x, 1×x=x, 1×6=6 → 6x²+37x+6 ✔️

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5. Factor: 2v² + 11v + 5

a×c = 2×5 = 10
Need two numbers that multiply to 10 and add to 11 → 10 and 1

Split:

2v² + 10v + v + 5

Group:

(2v² + 10v) + (v + 5)

Factor:

2v(v + 5) + 1(v + 5)

→ (2v + 1)(v + 5)

✔ Check: 2v×v=2v², 2v×5=10v, 1×v=v, 1×5=5 → 2v²+11v+5 ✔️

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6. Factor: 2b² + 17b + 21

a×c = 2×21 = 42
Need two numbers that multiply to 42 and add to 17 → 14 and 3

Split:

2b² + 14b + 3b + 21

Group:

(2b² + 14b) + (3b + 21)

Factor:

2b(b + 7) + 3(b + 7)

→ (2b + 3)(b + 7)

✔ Check: 2b×b=2b², 2b×7=14b, 3×b=3b, 3×7=21 → 2b²+17b+21 ✔️

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7. Factor: 7a² + 53a + 28

a×c = 7×28 = 196
Need two numbers that multiply to 196 and add to 53

Let’s list factor pairs of 196:

1 & 196 → sum 197
2 & 98 → 100
4 & 49 → 53 ← YES!

So split:

7a² + 49a + 4a + 28

Group:

(7a² + 49a) + (4a + 28)

Factor:

7a(a + 7) + 4(a + 7)

→ (7a + 4)(a + 7)

✔ Check: 7a×a=7a², 7a×7=49a, 4×a=4a, 4×7=28 → 7a²+53a+28 ✔️

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8. Factor: 9p² + 73p + 70

a×c = 9×70 = 630
Need two numbers that multiply to 630 and add to 73

This is trickier. Let’s think:

What two numbers near √630 ≈ 25? Try larger ones.

Try 63 and 10 → 63×10=630, 63+10=73 ← PERFECT!

Split:

9p² + 63p + 10p + 70

Group:

(9p² + 63p) + (10p + 70)

Factor:

9p(p + 7) + 10(p + 7)

→ (9p + 10)(p + 7)

✔ Check: 9p×p=9p², 9p×7=63p, 10×p=10p, 10×7=70 → 9p²+73p+70 ✔️

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Final Answer:
(5n + 4)(n + 3),
3(3k + 1)(k + 7),
(2n + 1)(n + 2),
(6x + 1)(x + 6),
(2v + 1)(v + 5),
(2b + 3)(b + 7),
(7a + 4)(a + 7),
(9p + 10)(p + 7)