Worksheet for practicing factorising quadratic equations, including problems and solutions space.
Factorising Quadratic Equations Sheet 11 worksheet with 16 quadratic equations to factorise, featuring a table format and a math-themed logo.
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Show Answer Key & Explanations
Step-by-step solution for: Factoring Quadratic Equations
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Show Answer Key & Explanations
Step-by-step solution for: Factoring Quadratic Equations
Let’s solve each quadratic equation by factorising. We’ll go one by one, step by step.
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1) 2x² + 13x + 15 = 0
We need to factorise into (2x + __)(x + __)
Look for two numbers that multiply to 2×15 = 30 and add to 13 → 10 and 3
So:
2x² + 10x + 3x + 15 = 0
Group: (2x² + 10x) + (3x + 15) = 0
Factor: 2x(x + 5) + 3(x + 5) = 0
→ (2x + 3)(x + 5) = 0
Set each bracket to zero:
2x + 3 = 0 → x = -3/2
x + 5 = 0 → x = -5
Answer: (2x + 3)(x + 5) and x = -3/2 or -5
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2) 2y² – y – 21 = 0
Multiply 2 × (-21) = -42
Find two numbers that multiply to -42 and add to -1 → -7 and 6
Split middle term:
2y² – 7y + 6y – 21 = 0
Group: (2y² – 7y) + (6y – 21) = 0
Factor: y(2y – 7) + 3(2y – 7) = 0
→ (y + 3)(2y – 7) = 0
Solutions:
y + 3 = 0 → y = -3
2y – 7 = 0 → y = 7/2
Answer: (y + 3)(2y – 7) and y = -3 or 7/2
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3) 2z² – 12z – 14 = 0
First, factor out common factor 2:
2(z² – 6z – 7) = 0
Now factor z² – 6z – 7 → find two numbers that multiply to -7 and add to -6 → -7 and 1
So: 2(z – 7)(z + 1) = 0
Solutions:
z – 7 = 0 → z = 7
z + 1 = 0 → z = -1
Answer: 2(z – 7)(z + 1) and z = 7 or -1
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4) 2a² – 21a – 11 = 0
Multiply 2 × (-11) = -22
Find two numbers that multiply to -22 and add to -21 → -22 and 1
Split: 2a² – 22a + a – 11 = 0
Group: (2a² – 22a) + (a – 11) = 0
Factor: 2a(a – 11) + 1(a – 11) = 0
→ (2a + 1)(a – 11) = 0
Solutions:
2a + 1 = 0 → a = -1/2
a – 11 = 0 → a = 11
Answer: (2a + 1)(a – 11) and a = -1/2 or 11
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5) 2b² + 10b + 12 = 0
Factor out 2 first: 2(b² + 5b + 6) = 0
Factor b² + 5b + 6 → (b + 2)(b + 3)
So: 2(b + 2)(b + 3) = 0
Solutions:
b + 2 = 0 → b = -2
b + 3 = 0 → b = -3
Answer: 2(b + 2)(b + 3) and b = -2 or -3
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6) 2c² – 13c + 6 = 0
Multiply 2 × 6 = 12
Find two numbers that multiply to 12 and add to -13 → -12 and -1
Split: 2c² – 12c – c + 6 = 0
Group: (2c² – 12c) + (-c + 6) = 0
Factor: 2c(c – 6) –1(c – 6) = 0
→ (2c – 1)(c – 6) = 0
Solutions:
2c – 1 = 0 → c = 1/2
c – 6 = 0 → c = 6
Answer: (2c – 1)(c – 6) and c = 1/2 or 6
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7) 2d² – 12d + 10 = 0
Factor out 2: 2(d² – 6d + 5) = 0
Factor d² – 6d + 5 → (d – 1)(d – 5)
So: 2(d – 1)(d – 5) = 0
Solutions:
d – 1 = 0 → d = 1
d – 5 = 0 → d = 5
Answer: 2(d – 1)(d – 5) and d = 1 or 5
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8) 2e² – 12e + 16 = 0
Factor out 2: 2(e² – 6e + 8) = 0
Factor e² – 6e + 8 → (e – 2)(e – 4)
So: 2(e – 2)(e – 4) = 0
Solutions:
e – 2 = 0 → e = 2
e – 4 = 0 → e = 4
Answer: 2(e – 2)(e – 4) and e = 2 or 4
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9) 3f² – 8f + 5 = 0
Multiply 3 × 5 = 15
Find two numbers that multiply to 15 and add to -8 → -5 and -3
Split: 3f² – 5f – 3f + 5 = 0
Group: (3f² – 5f) + (-3f + 5) = 0
Factor: f(3f – 5) –1(3f – 5) = 0
→ (f – 1)(3f – 5) = 0
Solutions:
f – 1 = 0 → f = 1
3f – 5 = 0 → f = 5/3
Answer: (f – 1)(3f – 5) and f = 1 or 5/3
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10) 3g² – 8g – 11 = 0
Multiply 3 × (-11) = -33
Find two numbers that multiply to -33 and add to -8 → -11 and 3
Split: 3g² – 11g + 3g – 11 = 0
Group: (3g² – 11g) + (3g – 11) = 0
Factor: g(3g – 11) + 1(3g – 11) = 0
→ (g + 1)(3g – 11) = 0
Solutions:
g + 1 = 0 → g = -1
3g – 11 = 0 → g = 11/3
Answer: (g + 1)(3g – 11) and g = -1 or 11/3
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11) 3h² + 10h + 7 = 0
Multiply 3 × 7 = 21
Find two numbers that multiply to 21 and add to 10 → 7 and 3
Split: 3h² + 7h + 3h + 7 = 0
Group: (3h² + 7h) + (3h + 7) = 0
Factor: h(3h + 7) + 1(3h + 7) = 0
→ (h + 1)(3h + 7) = 0
Solutions:
h + 1 = 0 → h = -1
3h + 7 = 0 → h = -7/3
Answer: (h + 1)(3h + 7) and h = -1 or -7/3
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12) 3i² – 8i – 3 = 0
Multiply 3 × (-3) = -9
Find two numbers that multiply to -9 and add to -8 → -9 and 1
Split: 3i² – 9i + i – 3 = 0
Group: (3i² – 9i) + (i – 3) = 0
Factor: 3i(i – 3) + 1(i – 3) = 0
→ (3i + 1)(i – 3) = 0
Solutions:
3i + 1 = 0 → i = -1/3
i – 3 = 0 → i = 3
Answer: (3i + 1)(i – 3) and i = -1/3 or 3
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13) 3j² – 25j + 8 = 0
Multiply 3 × 8 = 24
Find two numbers that multiply to 24 and add to -25 → -24 and -1
Split: 3j² – 24j – j + 8 = 0
Group: (3j² – 24j) + (-j + 8) = 0
Factor: 3j(j – 8) –1(j – 8) = 0
→ (3j – 1)(j – 8) = 0
Solutions:
3j – 1 = 0 → j = 1/3
j – 8 = 0 → j = 8
Answer: (3j – 1)(j – 8) and j = 1/3 or 8
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14) 5k² + 11k + 2 = 0
Multiply 5 × 2 = 10
Find two numbers that multiply to 10 and add to 11 → 10 and 1
Split: 5k² + 10k + k + 2 = 0
Group: (5k² + 10k) + (k + 2) = 0
Factor: 5k(k + 2) + 1(k + 2) = 0
→ (5k + 1)(k + 2) = 0
Solutions:
5k + 1 = 0 → k = -1/5
k + 2 = 0 → k = -2
Answer: (5k + 1)(k + 2) and k = -1/5 or -2
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15) 5m² – 6m – 11 = 0
Multiply 5 × (-11) = -55
Find two numbers that multiply to -55 and add to -6 → -11 and 5
Split: 5m² – 11m + 5m – 11 = 0
Group: (5m² – 11m) + (5m – 11) = 0
Factor: m(5m – 11) + 1(5m – 11) = 0
→ (m + 1)(5m – 11) = 0
Solutions:
m + 1 = 0 → m = -1
5m – 11 = 0 → m = 11/5
Answer: (m + 1)(5m – 11) and m = -1 or 11/5
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16) 5n² – 36n + 7 = 0
Multiply 5 × 7 = 35
Find two numbers that multiply to 35 and add to -36 → -35 and -1
Split: 5n² – 35n – n + 7 = 0
Group: (5n² – 35n) + (-n + 7) = 0
Factor: 5n(n – 7) –1(n – 7) = 0
→ (5n – 1)(n – 7) = 0
Solutions:
5n – 1 = 0 → n = 1/5
n – 7 = 0 → n = 7
Answer: (5n – 1)(n – 7) and n = 1/5 or 7
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Final Answer:
1) (2x + 3)(x + 5); x = -3/2 or -5
2) (y + 3)(2y – 7); y = -3 or 7/2
3) 2(z – 7)(z + 1); z = 7 or -1
4) (2a + 1)(a – 11); a = -1/2 or 11
5) 2(b + 2)(b + 3); b = -2 or -3
6) (2c – 1)(c – 6); c = 1/2 or 6
7) 2(d – 1)(d – 5); d = 1 or 5
8) 2(e – 2)(e – 4); e = 2 or 4
9) (f – 1)(3f – 5); f = 1 or 5/3
10) (g + 1)(3g – 11); g = -1 or 11/3
11) (h + 1)(3h + 7); h = -1 or -7/3
12) (3i + 1)(i – 3); i = -1/3 or 3
13) (3j – 1)(j – 8); j = 1/3 or 8
14) (5k + 1)(k + 2); k = -1/5 or -2
15) (m + 1)(5m – 11); m = -1 or 11/5
16) (5n – 1)(n – 7); n = 1/5 or 7
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1) 2x² + 13x + 15 = 0
We need to factorise into (2x + __)(x + __)
Look for two numbers that multiply to 2×15 = 30 and add to 13 → 10 and 3
So:
2x² + 10x + 3x + 15 = 0
Group: (2x² + 10x) + (3x + 15) = 0
Factor: 2x(x + 5) + 3(x + 5) = 0
→ (2x + 3)(x + 5) = 0
Set each bracket to zero:
2x + 3 = 0 → x = -3/2
x + 5 = 0 → x = -5
Answer: (2x + 3)(x + 5) and x = -3/2 or -5
---
2) 2y² – y – 21 = 0
Multiply 2 × (-21) = -42
Find two numbers that multiply to -42 and add to -1 → -7 and 6
Split middle term:
2y² – 7y + 6y – 21 = 0
Group: (2y² – 7y) + (6y – 21) = 0
Factor: y(2y – 7) + 3(2y – 7) = 0
→ (y + 3)(2y – 7) = 0
Solutions:
y + 3 = 0 → y = -3
2y – 7 = 0 → y = 7/2
Answer: (y + 3)(2y – 7) and y = -3 or 7/2
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3) 2z² – 12z – 14 = 0
First, factor out common factor 2:
2(z² – 6z – 7) = 0
Now factor z² – 6z – 7 → find two numbers that multiply to -7 and add to -6 → -7 and 1
So: 2(z – 7)(z + 1) = 0
Solutions:
z – 7 = 0 → z = 7
z + 1 = 0 → z = -1
Answer: 2(z – 7)(z + 1) and z = 7 or -1
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4) 2a² – 21a – 11 = 0
Multiply 2 × (-11) = -22
Find two numbers that multiply to -22 and add to -21 → -22 and 1
Split: 2a² – 22a + a – 11 = 0
Group: (2a² – 22a) + (a – 11) = 0
Factor: 2a(a – 11) + 1(a – 11) = 0
→ (2a + 1)(a – 11) = 0
Solutions:
2a + 1 = 0 → a = -1/2
a – 11 = 0 → a = 11
Answer: (2a + 1)(a – 11) and a = -1/2 or 11
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5) 2b² + 10b + 12 = 0
Factor out 2 first: 2(b² + 5b + 6) = 0
Factor b² + 5b + 6 → (b + 2)(b + 3)
So: 2(b + 2)(b + 3) = 0
Solutions:
b + 2 = 0 → b = -2
b + 3 = 0 → b = -3
Answer: 2(b + 2)(b + 3) and b = -2 or -3
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6) 2c² – 13c + 6 = 0
Multiply 2 × 6 = 12
Find two numbers that multiply to 12 and add to -13 → -12 and -1
Split: 2c² – 12c – c + 6 = 0
Group: (2c² – 12c) + (-c + 6) = 0
Factor: 2c(c – 6) –1(c – 6) = 0
→ (2c – 1)(c – 6) = 0
Solutions:
2c – 1 = 0 → c = 1/2
c – 6 = 0 → c = 6
Answer: (2c – 1)(c – 6) and c = 1/2 or 6
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7) 2d² – 12d + 10 = 0
Factor out 2: 2(d² – 6d + 5) = 0
Factor d² – 6d + 5 → (d – 1)(d – 5)
So: 2(d – 1)(d – 5) = 0
Solutions:
d – 1 = 0 → d = 1
d – 5 = 0 → d = 5
Answer: 2(d – 1)(d – 5) and d = 1 or 5
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8) 2e² – 12e + 16 = 0
Factor out 2: 2(e² – 6e + 8) = 0
Factor e² – 6e + 8 → (e – 2)(e – 4)
So: 2(e – 2)(e – 4) = 0
Solutions:
e – 2 = 0 → e = 2
e – 4 = 0 → e = 4
Answer: 2(e – 2)(e – 4) and e = 2 or 4
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9) 3f² – 8f + 5 = 0
Multiply 3 × 5 = 15
Find two numbers that multiply to 15 and add to -8 → -5 and -3
Split: 3f² – 5f – 3f + 5 = 0
Group: (3f² – 5f) + (-3f + 5) = 0
Factor: f(3f – 5) –1(3f – 5) = 0
→ (f – 1)(3f – 5) = 0
Solutions:
f – 1 = 0 → f = 1
3f – 5 = 0 → f = 5/3
Answer: (f – 1)(3f – 5) and f = 1 or 5/3
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10) 3g² – 8g – 11 = 0
Multiply 3 × (-11) = -33
Find two numbers that multiply to -33 and add to -8 → -11 and 3
Split: 3g² – 11g + 3g – 11 = 0
Group: (3g² – 11g) + (3g – 11) = 0
Factor: g(3g – 11) + 1(3g – 11) = 0
→ (g + 1)(3g – 11) = 0
Solutions:
g + 1 = 0 → g = -1
3g – 11 = 0 → g = 11/3
Answer: (g + 1)(3g – 11) and g = -1 or 11/3
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11) 3h² + 10h + 7 = 0
Multiply 3 × 7 = 21
Find two numbers that multiply to 21 and add to 10 → 7 and 3
Split: 3h² + 7h + 3h + 7 = 0
Group: (3h² + 7h) + (3h + 7) = 0
Factor: h(3h + 7) + 1(3h + 7) = 0
→ (h + 1)(3h + 7) = 0
Solutions:
h + 1 = 0 → h = -1
3h + 7 = 0 → h = -7/3
Answer: (h + 1)(3h + 7) and h = -1 or -7/3
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12) 3i² – 8i – 3 = 0
Multiply 3 × (-3) = -9
Find two numbers that multiply to -9 and add to -8 → -9 and 1
Split: 3i² – 9i + i – 3 = 0
Group: (3i² – 9i) + (i – 3) = 0
Factor: 3i(i – 3) + 1(i – 3) = 0
→ (3i + 1)(i – 3) = 0
Solutions:
3i + 1 = 0 → i = -1/3
i – 3 = 0 → i = 3
Answer: (3i + 1)(i – 3) and i = -1/3 or 3
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13) 3j² – 25j + 8 = 0
Multiply 3 × 8 = 24
Find two numbers that multiply to 24 and add to -25 → -24 and -1
Split: 3j² – 24j – j + 8 = 0
Group: (3j² – 24j) + (-j + 8) = 0
Factor: 3j(j – 8) –1(j – 8) = 0
→ (3j – 1)(j – 8) = 0
Solutions:
3j – 1 = 0 → j = 1/3
j – 8 = 0 → j = 8
Answer: (3j – 1)(j – 8) and j = 1/3 or 8
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14) 5k² + 11k + 2 = 0
Multiply 5 × 2 = 10
Find two numbers that multiply to 10 and add to 11 → 10 and 1
Split: 5k² + 10k + k + 2 = 0
Group: (5k² + 10k) + (k + 2) = 0
Factor: 5k(k + 2) + 1(k + 2) = 0
→ (5k + 1)(k + 2) = 0
Solutions:
5k + 1 = 0 → k = -1/5
k + 2 = 0 → k = -2
Answer: (5k + 1)(k + 2) and k = -1/5 or -2
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15) 5m² – 6m – 11 = 0
Multiply 5 × (-11) = -55
Find two numbers that multiply to -55 and add to -6 → -11 and 5
Split: 5m² – 11m + 5m – 11 = 0
Group: (5m² – 11m) + (5m – 11) = 0
Factor: m(5m – 11) + 1(5m – 11) = 0
→ (m + 1)(5m – 11) = 0
Solutions:
m + 1 = 0 → m = -1
5m – 11 = 0 → m = 11/5
Answer: (m + 1)(5m – 11) and m = -1 or 11/5
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16) 5n² – 36n + 7 = 0
Multiply 5 × 7 = 35
Find two numbers that multiply to 35 and add to -36 → -35 and -1
Split: 5n² – 35n – n + 7 = 0
Group: (5n² – 35n) + (-n + 7) = 0
Factor: 5n(n – 7) –1(n – 7) = 0
→ (5n – 1)(n – 7) = 0
Solutions:
5n – 1 = 0 → n = 1/5
n – 7 = 0 → n = 7
Answer: (5n – 1)(n – 7) and n = 1/5 or 7
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Final Answer:
1) (2x + 3)(x + 5); x = -3/2 or -5
2) (y + 3)(2y – 7); y = -3 or 7/2
3) 2(z – 7)(z + 1); z = 7 or -1
4) (2a + 1)(a – 11); a = -1/2 or 11
5) 2(b + 2)(b + 3); b = -2 or -3
6) (2c – 1)(c – 6); c = 1/2 or 6
7) 2(d – 1)(d – 5); d = 1 or 5
8) 2(e – 2)(e – 4); e = 2 or 4
9) (f – 1)(3f – 5); f = 1 or 5/3
10) (g + 1)(3g – 11); g = -1 or 11/3
11) (h + 1)(3h + 7); h = -1 or -7/3
12) (3i + 1)(i – 3); i = -1/3 or 3
13) (3j – 1)(j – 8); j = 1/3 or 8
14) (5k + 1)(k + 2); k = -1/5 or -2
15) (m + 1)(5m – 11); m = -1 or 11/5
16) (5n – 1)(n – 7); n = 1/5 or 7
Parent Tip: Review the logic above to help your child master the concept of solving quadratic equations by formula worksheet.
