Factor quadratics worksheet with expressions like 5n² + 19n + 12 and 9k² + 66k + 21.

To factor these quadratic expressions, we’ll use the “ac method” — a reliable way to break down quadratics when the coefficient of the squared term (a) is greater than 1.

Here’s how it works:
- Multiply a and c (the first and last numbers in the trinomial).
- Find two numbers that multiply to give you that product, and add up to b (the middle number).
- Use those two numbers to split the middle term.
- Then factor by grouping.

Let’s go one by one.

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1. 5n² + 19n + 12

a = 5, b = 19, c = 12
ac = 5 × 12 = 60
Find two numbers that multiply to 60 and add to 19 → 15 and 4

Split the middle term:
5n² + 15n + 4n + 12

Group:
(5n² + 15n) + (4n + 12)
= 5n(n + 3) + 4(n + 3)
= (5n + 4)(n + 3)

✔ Check: (5n+4)(n+3) = 5n² + 15n + 4n + 12 = 5n² + 19n + 12 ✔️

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2. 9k² + 66k + 21

First, check if there’s a common factor. All terms are divisible by 3!

Factor out 3 first:
3(3k² + 22k + 7)

Now factor inside:
a = 3, b = 22, c = 7
ac = 21
Numbers that multiply to 21 and add to 22 → 21 and 1

Split:
3k² + 21k + k + 7
Group:
(3k² + 21k) + (k + 7)
= 3k(k + 7) + 1(k + 7)
= (3k + 1)(k + 7)

Don’t forget the 3 we factored out earlier:
→ 3(3k + 1)(k + 7)

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

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3. 2n² + 5n + 2

a = 2, b = 5, c = 2
ac = 4
Numbers that multiply to 4 and add to 5 → 4 and 1

Split:
2n² + 4n + n + 2
Group:
(2n² + 4n) + (n + 2)
= 2n(n + 2) + 1(n + 2)
= (2n + 1)(n + 2)

✔ Check: (2n+1)(n+2) = 2n² + 4n + n + 2 = 2n² + 5n + 2 ✔️

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4. 6x² + 37x + 6

a = 6, b = 37, c = 6
ac = 36
Numbers that multiply to 36 and add to 37 → 36 and 1

Split:
6x² + 36x + x + 6
Group:
(6x² + 36x) + (x + 6)
= 6x(x + 6) + 1(x + 6)
= (6x + 1)(x + 6)

✔ Check: (6x+1)(x+6) = 6x² + 36x + x + 6 = 6x² + 37x + 6 ✔️

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5. 2v² + 11v + 5

a = 2, b = 11, c = 5
ac = 10
Numbers that multiply to 10 and add to 11 → 10 and 1

Split:
2v² + 10v + v + 5
Group:
(2v² + 10v) + (v + 5)
= 2v(v + 5) + 1(v + 5)
= (2v + 1)(v + 5)

✔ Check: (2v+1)(v+5) = 2v² + 10v + v + 5 = 2v² + 11v + 5 ✔️

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6. 2b² + 17b + 21

a = 2, b = 17, c = 21
ac = 42
Numbers that multiply to 42 and add to 17 → 14 and 3

Split:
2b² + 14b + 3b + 21
Group:
(2b² + 14b) + (3b + 21)
= 2b(b + 7) + 3(b + 7)
= (2b + 3)(b + 7)

✔ Check: (2b+3)(b+7) = 2b² + 14b + 3b + 21 = 2b² + 17b + 21 ✔️

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7. 7a² + 53a + 28

a = 7, b = 53, c = 28
ac = 196
Need two numbers that multiply to 196 and add to 53.

Let’s list factors of 196:
1 & 196 → sum 197
2 & 98 → 100
4 & 49 → 53 ← YES!

Split:
7a² + 49a + 4a + 28
Group:
(7a² + 49a) + (4a + 28)
= 7a(a + 7) + 4(a + 7)
= (7a + 4)(a + 7)

✔ Check: (7a+4)(a+7) = 7a² + 49a + 4a + 28 = 7a² + 53a + 28 ✔️

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8. 9p² + 73p + 70

a = 9, b = 73, c = 70
ac = 630
Need two numbers that multiply to 630 and add to 73.

Try pairs:
630 ÷ 10 = 63 → 10 + 63 = 73 ← YES!

Split:
9p² + 63p + 10p + 70
Group:
(9p² + 63p) + (10p + 70)
= 9p(p + 7) + 10(p + 7)
= (9p + 10)(p + 7)

✔ Check: (9p+10)(p+7) = 9p² + 63p + 10p + 70 = 9p² + 73p + 70 ✔️

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Final Answer:
5n² + 19n + 12 = (5n + 4)(n + 3)
9k² + 66k + 21 = 3(3k + 1)(k + 7)
2n² + 5n + 2 = (2n + 1)(n + 2)
6x² + 37x + 6 = (6x + 1)(x + 6)
2v² + 11v + 5 = (2v + 1)(v + 5)
2b² + 17b + 21 = (2b + 3)(b + 7)
7a² + 53a + 28 = (7a + 4)(a + 7)
9p² + 73p + 70 = (9p + 10)(p + 7)