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Factorising harder quadratics

Free Printable Factorising harder quadratics PDF Worksheets, Revision Guide & GCSE Past Paper Questions (Grade 7)

Download free factorising harder quadratics worksheets (PDF), GCSE maths past paper questions and worked solutions for Factorising harder quadratics (7F). This is a Grade 7 algebra topic from the GCSE questions by topic collection for Higher tier GCSE Maths (Edexcel, AQA, OCR and IGCSE).

The free PDF worksheets table below brings together printable question booklets (Qs) and worked answers or mark schemes (Ans) from Maths Genie, Corbett Maths, Maths Made Easy, Maths DIY, Dr Austin, E Classroom and PMT, along with past paper question booklets.

Below the worksheet links you can read our factorising harder quadratics revision notes and worked examples with diagrams, try real GCSE past paper factorising harder quadratics questions with step-by-step solutions, or download a PDF of all factorising harder quadratics worksheet links.

Free Factorising harder quadratics PDF Worksheets & Answers

Grade 7 • Topic 7F
Resource Questions Answers
Graded exam questions & solutions
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Practice Questions & answers
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Textbook Exercise & answers
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MME worksheet & model answers
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Exam-style booklet & solutions
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GCSE topic booklet & answers
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PMT (Edxl)
Edexcel GCSE topic questions & solutions
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PMT (AQA)
AQA GCSE topic questions & solutions
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PMT (IGCSE)
Edexcel IGCSE topic questions & solutions
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Online topic notes & examples
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Grade 7 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
GCSE Maths Grade 7 Revision Notes & Method

Guide to Factorising harder quadratics: Explanation, Diagrams & Worked Examples

Download Printable One Page Guide (PDF)

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
Scanned GCSE Exam Questions & Worked Solutions

GCSE Past Paper Factorising harder quadratics Questions & Answers

Past Paper Example 1 IGCSE Jun 2022 1H Q11 5 marks
Full Paper (PDF) • Mark Scheme (PDF)
(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 & Mark Scheme (5 marks)
Part (a) Solution:
(3x)² – (2y)² = (3x + 2y)(3x – 2y) M1 A1
Answer (a): (3x + 2y)(3x - 2y)
Part (b) Solution:
7x/(8x) – 2(x + 3)/(8x) = (7x – 2x – 6)/(8x) = (5x – 6)/(8x) M2 A1
Answer (b): 5x - 68x
Past Paper Example 2 GCSE Nov 2017 2F Q24 5 marks
Full Paper (PDF) • Mark Scheme (PDF)
(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 & Mark Scheme (5 marks)
Part (a) Solution:
x² = 36 M1
⇒ x = ±6 A1
Answer (a): ± 6 (or 6, -6)
Part (b) Solution:
6x² – 4x + 3x – 2 = 6x² – x – 2 M1 A1
Answer (b): 6x2 - x - 2
Part (c) Solution:
(x + 3)(x + 3) = (x + 3)² B1
Answer (c): (x + 3)2
Past Paper Example 3 AQA Nov 2024 3H Q20 2 marks
Full Paper (PDF) • Mark Scheme (PDF)
Factorise fully 3x2 + 23x + 30
Worked Solution & Mark Scheme (2 marks)
3x² + 18x + 5x + 30 = 3x(x + 6) + 5(x + 6) = (3x + 5)(x + 6). M1 A1
Answer: Answer (3x + 5)(x + 6)

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