Quadratic Equations Worksheets - Free Printable

Let’s solve each quadratic equation by factoring. We’ll go one at a time, step by step.

---

1. 4d² + 30d + 50 = 0

First, factor out the greatest common factor (GCF). All terms are divisible by 2:

→ 2(2d² + 15d + 25) = 0

Now factor inside the parentheses:
We need two numbers that multiply to 2×25 = 50 and add to 15 → those are 10 and 5.

So:
2d² + 10d + 5d + 25 = 0
Group: (2d² + 10d) + (5d + 25) = 0
Factor: 2d(d + 5) + 5(d + 5) = 0
→ (2d + 5)(d + 5) = 0

Set each factor to zero:
2d + 5 = 0 → d = -5/2
d + 5 = 0 → d = -5

✔ Answer: d = {-5/2, -5} — matches given answer.

---

2. 3x² + 11x + 6 = 0

Multiply leading coefficient × constant: 3 × 6 = 18
Find two numbers that multiply to 18 and add to 11 → 9 and 2.

Rewrite middle term:
3x² + 9x + 2x + 6 = 0
Group: (3x² + 9x) + (2x + 6) = 0
Factor: 3x(x + 3) + 2(x + 3) = 0
→ (3x + 2)(x + 3) = 0

Set each factor to zero:
3x + 2 = 0 → x = -2/3
x + 3 = 0 → x = -3

✔ Answer: x = {-2/3, -3}

---

3. 3v² + 23v - 36 = 0

Multiply 3 × (-36) = -108
Need two numbers that multiply to -108 and add to 23 → 27 and -4.

Rewrite:
3v² + 27v - 4v - 36 = 0
Group: (3v² + 27v) + (-4v - 36) = 0
Factor: 3v(v + 9) -4(v + 9) = 0
→ (3v - 4)(v + 9) = 0

Set each factor to zero:
3v - 4 = 0 → v = 4/3
v + 9 = 0 → v = -9

✔ Answer: v = {4/3, -9}

---

4. 2q² + 18q + 40 = 0

Factor out GCF first: all divisible by 2 →
2(q² + 9q + 20) = 0

Factor inside: q² + 9q + 20 → factors of 20 that add to 9 → 4 and 5
→ (q + 4)(q + 5) = 0

Set each factor to zero:
q + 4 = 0 → q = -4
q + 5 = 0 → q = -5

✔ Answer: q = {-4, -5}

---

5. 5r² + 27r + 28 = 0

Multiply 5 × 28 = 140
Need two numbers that multiply to 140 and add to 27 → 20 and 7.

Rewrite:
5r² + 20r + 7r + 28 = 0
Group: (5r² + 20r) + (7r + 28) = 0
Factor: 5r(r + 4) + 7(r + 4) = 0
→ (5r + 7)(r + 4) = 0

Set each factor to zero:
5r + 7 = 0 → r = -7/5
r + 4 = 0 → r = -4

✔ Answer: r = {-7/5, -4}

---

6. 4b² + 7b - 15 = 0

Multiply 4 × (-15) = -60
Need two numbers that multiply to -60 and add to 7 → 12 and -5.

Rewrite:
4b² + 12b - 5b - 15 = 0
Group: (4b² + 12b) + (-5b - 15) = 0
Factor: 4b(b + 3) -5(b + 3) = 0
→ (4b - 5)(b + 3) = 0

Set each factor to zero:
4b - 5 = 0 → b = 5/4
b + 3 = 0 → b = -3

✔ Answer: b = {5/4, -3}

---

7. 9y² + 9y - 4 = 0

Multiply 9 × (-4) = -36
Need two numbers that multiply to -36 and add to 9 → 12 and -3.

Rewrite:
9y² + 12y - 3y - 4 = 0
Group: (9y² + 12y) + (-3y - 4) = 0
Factor: 3y(3y + 4) -1(3y + 4) = 0
→ (3y - 1)(3y + 4) = 0

Wait — let me check that again.

Actually:
After grouping:
3y(3y + 4) -1(3y + 4) → yes, so (3y - 1)(3y + 4) = 0? Let's expand to verify:

(3y - 1)(3y + 4) = 9y² + 12y - 3y - 4 = 9y² + 9y - 4 ✔ Correct.

Set each factor to zero:
3y - 1 = 0 → y = 1/3
3y + 4 = 0 → y = -4/3

But wait — the answer key says y = {¹⁄₃, -⁴⁄₃} — which is same as {1/3, -4/3}. So correct.

✔ Answer: y = {1/3, -4/3}

---

8. 6k² - 11k - 35 = 0

Multiply 6 × (-35) = -210
Need two numbers that multiply to -210 and add to -11 → -21 and 10.

Rewrite:
6k² - 21k + 10k - 35 = 0
Group: (6k² - 21k) + (10k - 35) = 0
Factor: 3k(2k - 7) + 5(2k - 7) = 0
→ (3k + 5)(2k - 7) = 0

Set each factor to zero:
3k + 5 = 0 → k = -5/3
2k - 7 = 0 → k = 7/2

✔ Answer: k = {-5/3, 7/2}

---

9. 8p² - 2p - 28 = 0

Factor out GCF: all divisible by 2 →
2(4p² - p - 14) = 0

Now factor 4p² - p - 14
Multiply 4 × (-14) = -56
Need two numbers that multiply to -56 and add to -1 → -8 and 7.

Rewrite:
4p² - 8p + 7p - 14 = 0
Group: (4p² - 8p) + (7p - 14) = 0
Factor: 4p(p - 2) + 7(p - 2) = 0
→ (4p + 7)(p - 2) = 0

Set each factor to zero:
4p + 7 = 0 → p = -7/4
p - 2 = 0 → p = 2

✔ Answer: p = {-7/4, 2}

---

10. 2a² + 27a + 81 = 0

Multiply 2 × 81 = 162
Need two numbers that multiply to 162 and add to 27 → 18 and 9.

Rewrite:
2a² + 18a + 9a + 81 = 0
Group: (2a² + 18a) + (9a + 81) = 0
Factor: 2a(a + 9) + 9(a + 9) = 0
→ (2a + 9)(a + 9) = 0

Set each factor to zero:
2a + 9 = 0 → a = -9/2
a + 9 = 0 → a = -9

✔ Answer: a = {-9/2, -9}

---

Final Answer:
All answers match the provided answer key. Here they are listed clearly:

- d = {-5/2, -5}
- x = {-2/3, -3}
- v = {4/3, -9}
- q = {-4, -5}
- r = {-7/5, -4}
- b = {5/4, -3}
- y = {1/3, -4/3}
- k = {-5/3, 7/2}
- p = {-7/4, 2}
- a = {-9/2, -9}