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Factorising harder quadratics Quiz

Grade 7 • 7F • 10 Questions (41 Marks)

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Factorising harder quadratics — Notes & Worked Examples Show / Hide

Explanation:

To factorise a harder quadratic ax2 + bx + c (where a > 1) by splitting the middle term, multiply a × c first and find two numbers that multiply to give ac and add to give b. Rewrite the middle term bx using these two numbers, then factorise in pairs by grouping.

Always check first if there is a common numerical factor that can be taken out of all three terms, or if the expression is a difference of two squares such as 9x2 − 16y2 = (3x − 4y)(3x + 4y). If solving an equation equal to zero, set each bracket equal to zero.

Example 1: Factorise 6x2 − 11x − 10

Solution:

ac = 6 × (−10) = −60; b = −11
Numbers multiplying to −60 and adding to −11 are −15 and +4
Split middle term: 6x2 − 15x + 4x − 10
Factorise in pairs: 3x(2x − 5) + 2(2x − 5)
Answer: (3x + 2)(2x − 5)

Example 2: Solve 4x2 + 5x − 6 = 0

Solution:

ac = 4 × (−6) = −24; factors adding to +5 are +8 and −3
4x2 + 8x − 3x − 6 = 0
4x(x + 2) − 3(x + 2) = 0 ⇒ (4x − 3)(x + 2) = 0
Set each bracket to 0: 4x − 3 = 0 or x + 2 = 0
Answer: x = 34 or x = −2
Question 1 AQA Nov 2024 3H Q20
[2 marks]
Factorise fully 3x2 + 23x + 30
Answer
Worked Solution & Answer
3x² + 18x + 5x + 30 = 3x(x + 6) + 5(x + 6) = (3x + 5)(x + 6). M1 A1
Answer: Answer (3x + 5)(x + 6)
Question Paper (PDF) • Mark Scheme (PDF)
Question 2 AQA Jun 2022 2H Q23
[2 marks]
Factorise 3x2 - 16x - 12
Answer
Worked Solution & Answer
We seek two numbers with product 3 × (-12) = -36 and sum -16: M1
These numbers are -18 and +2
Rewrite and factorise by grouping:
3x2 - 18x + 2x - 12 = 3x(x - 6) + 2(x - 6) = (3x + 2)(x - 6) A1
Answer: Answer (3x + 2)(x - 6)
Question Paper (PDF) • Mark Scheme (PDF)
Question 3 AQA Nov 2020 2H Q25
[2 marks]
Factorise 3x2 + 11x - 20
Answer
Worked Solution & Answer
Two numbers with product 3 × (-20) = -60 and sum 11 are 15 and -4: M1
3x2 + 15x - 4x - 20 = 3x(x + 5) - 4(x + 5) = (3x - 4)(x + 5). A1
Answer: Answer (3x - 4)(x + 5)
Question Paper (PDF) • Mark Scheme (PDF)
Question 4
[3 marks]
Solve 6x2 + 5x - 6 = 0
Worked Solution & Answer
(2x + 3)(3x – 2) = 0 M2
⇒ x = –3/2 (or –1.5) and x = 2/3 A1
Answer: -32 and 23
Question Paper (PDF) • Mark Scheme (PDF)
Question 5 AQA Jun 2025 3H Q20
[3 marks]
(a) Factorise fully 3n² + 5n + 2. [2 marks]
20 (b) Are any terms of 3n² + 5n + 2 prime? Give a reason. [1 mark]
(a) Factorise fully 3n2 + 5n + 2 [2 marks]
Answer
(b) A sequence has nth term 3n2 + 5n + 2
Are any of the terms in the sequence a prime number?
Tick a box.
Yes [ ] No [ ]
Give a reason for your answer. [1 mark]
Worked Solution & Answer
(a)
3n² + 5n + 2 = (3n + 2)(n + 1) M1 A1
Answer (a): Answer (3n + 2)(n + 1)
(b)
For n ≥ 1, n + 1 ≥ 2 and 3n + 2 ≥ 5, so every term is the product of two integers greater than 1 C1
Answer (b): No — since n ≥ 1, both factors (3n + 2) and (n + 1) are integers greater than 1
Question Paper (PDF) • Mark Scheme (PDF)
Question 6
[4 marks]
(a) Factorise 6x² – 5x – 4 (2) (b) Hence, or otherwise, solve 6x² – 5x – 4 < 0 (2)
(a) Factorise 6x2 - 5x - 4 [2 marks]
(b) Hence, or otherwise, solve 6x2 - 5x - 4 < 0 [2 marks]
Worked Solution & Answer
(a)
6x² – 8x + 3x – 4 = 2x(3x – 4) + 1(3x – 4) = (2x + 1)(3x – 4) M1 A1
Answer (a): (2x + 1)(3x - 4)
(b)
Critical values x = –1/2 and x = 4/3 M1
⇒ –1/2 < x < 4/3 A1
Answer (b): -12 < x < 43
Question Paper (PDF) • Mark Scheme (PDF)
Question 7 IGCSE Jun 2022 1H Q11
[5 marks]
(a) Factorise 9x² – 4y². (2) (b) Express 7/8 – (x + 3)/(4x) as a single fraction in simplest form. (3)
(a) Factorise 9x2 - 4y2 [2 marks]
(b) Express 78 - x + 34x as a single fraction in its simplest form. [3 marks]
Worked Solution & Answer
(a)
(3x)² – (2y)² = (3x + 2y)(3x – 2y) M1 A1
Answer (a): (3x + 2y)(3x - 2y)
(b)
7x/(8x) – 2(x + 3)/(8x) = (7x – 2x – 6)/(8x) = (5x – 6)/(8x) M2 A1
Answer (b): 5x - 68x
Question Paper (PDF) • Mark Scheme (PDF)
Question 8 IGCSE Jun 2024 1H Q12
[5 marks]
(a) Factorise fully 6y² − 5y − 4 (2) (b) Express (2x + 1)/(4x) + (7 − 5x)/(3x) as a single fraction in its simplest form. (3)
(a) Factorise fully 6y2 - 5y - 4 [2 marks]
(b) Express 2x + 14x + 7 - 5x3x as a single fraction in its simplest form. [3 marks]
Worked Solution & Answer
(a)
6y² − 8y + 3y − 4 = 2y(3y − 4) + 1(3y − 4) = (2y + 1)(3y − 4) M1 A1
Answer (a): (2y + 1)(3y - 4)
(b)
[3(2x + 1) + 4(7 − 5x)] / (12x) = (6x + 3 + 28 − 20x) / (12x) = (31 − 14x) / (12x) M2 A1
Answer (b): 31 - 14x12x
Question Paper (PDF) • Mark Scheme (PDF)
Question 9
[5 marks]
(a) Solve 2x² = 72. (2) (b) Expand and simplify (2x + 1)(3x – 2). (2) (c) Factorise x² + 6x + 9. (1)
(a) Solve 2x2 = 72 [2 marks]
(b) Expand and simplify (2x + 1)(3x - 2) [2 marks]
(c) Factorise x2 + 6x + 9 [1 mark]
Worked Solution & Answer
(a)
x² = 36 M1
⇒ x = ±6 A1
Answer (a): ± 6 (or 6, -6)
(b)
6x² – 4x + 3x – 2 = 6x² – x – 2 M1 A1
Answer (b): 6x2 - x - 2
(c)
(x + 3)(x + 3) = (x + 3)² B1
Answer (c): (x + 3)2
Question Paper (PDF) • Mark Scheme (PDF)
Question 10 OCR Jun 2024 4H Q15
[10 marks]
(a) Solve by factorisation.
3x2 + 10x - 8 = 0
(b) Write x2 + 8x + 11 in the form (x + a)2 - b.
(c) (i) Write down the coordinates of the turning point of the graph y = (x - 3)2 + 8.
(ii) Describe the **single** transformation which maps the graph of y = x2 onto the graph of y = (x - 3)2 + 8.
(a) Solve by factorisation.
3x2 + 10x - 8 = 0 [3 marks]
x =
(b) Write x2 + 8x + 11 in the form (x + a)2 - b. [3 marks]
(c) Write down the coordinates of the turning point of the graph y = (x - 3)2 + 8. [2 marks]
(d) Describe the **single** transformation which maps the graph of y = x2 onto the graph of y = (x - 3)2 + 8. [2 marks]
Worked Solution & Answer
(a)
(3x − 2)(x + 4) = 0 M2
⇒ x = 2/3 or x = −4. A1
Answer (a): x = 23 or -4
(b)
x² + 8x + 11 = (x + 4)² − 16 + 11 = (x + 4)² − 5. M2 A1
Answer (b): (x + 4)2 - 5
(c)
Minimum of (x − 3)² + 8 occurs at x = 3, y = 8 M1
⇒ (3, 8). A1
Answer (c): (3, 8)
(d)
Replacing x with x − 3 and adding 8 translates the graph 3 units right and 8 units up: translation by vector (3, 8). M1 A1
Answer (d): Translation by vector (38)
Question Paper (PDF) • Mark Scheme (PDF)
Score: 0 / 10 (0 / 41 marks)
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Grade 7 Quizzes

19 Topics
7A - Surds7B - Bounds
7C - Proportion (harder)7D - Algebraic fractions
7E - The quadratic formula7F - Factorising harder quadratics
7G - Trigonometric & exp. Graphs7H - Rearranging (harder)
7J - Functions (harder)7K - Iteration
7L - Area of any triangle7M - The sine rule
7N - The cosine rule7P - Congruent triangles
7Q - 3D pythagoras / trig.7R - Histograms
7S - Conditional probability7T - Exact trig. values (harder)
7U - Area of a segment