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Quadratic inequalities

Free Printable Quadratic inequalities PDF Worksheets, Revision Guide & GCSE Past Paper Questions (Grade 8+)

Download free quadratic inequalities worksheets (PDF), GCSE maths past paper questions and worked solutions for Quadratic inequalities (8E). This is a Grade 8+ algebra topic from the GCSE questions by topic collection for Foundation and 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, Dr Frost, Naiker Maths, 1st Class Maths, Metatutor, Dr Austin, E Classroom and PMT, along with past paper question booklets.

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

Free Quadratic inequalities PDF Worksheets & Answers

Grade 8+ • Topic 8E
Resource Questions Answers
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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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GCSE worksheet & solutions
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Exam question booklet & solutions
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Edexcel exam questions & solutions
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AQA exam questions & 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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Interactive online lesson & practice
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Grade 8+ topics
8A - Quad. sim. equations8B - Transforming graphs y=f(x)
8C - Algebraic proof8D - Completing the square
8E - Quadratic inequalities8F - Quadratic sequences
8G - Velocity time graphs8H - Circles and tangents
8J - Circle theorem proof8K - Vector proof
8L - Probability equation questions8M - Invariant points
8N - Spheres, cones and cylinders8P - Volume of a frustum
GCSE Maths Grade 8+ Revision Notes & Method

Guide to Quadratic inequalities: Explanation, Diagrams & Worked Examples

Download Printable One Page Guide (PDF)

Explanation:

Never solve a quadratic inequality like a linear equation: always rearrange so one side is 0 (with a positive x2 term), factorise to find the critical values (roots), and sketch a quick U-shaped parabola crossing the x-axis at those roots.

• If the quadratic is < 0 (below the x-axis), the solution is a single connected region between the roots: a < x < b.
• If the quadratic is > 0 (above the x-axis), the solution is two separate regions outside the roots: x < a or x > b.

Example 1: Solve x2 − 4x − 12 < 0

−2 6 −2 < x < 6

Solution:

Factorise: (x − 6)(x + 2) < 0
Critical values (roots): x = −2 and x = 6
U-shaped curve is below the x-axis (< 0) between the roots:
Answer: −2 < x < 6

Example 2: Solve 3x2 ≥ 10x + 8

−2/3 4

Solution:

Rearrange & factorise: 3x2 − 10x − 8 = (3x + 2)(x − 4) ≥ 0
Critical values: x = −23 and x = 4
Curve is on or above x-axis (≥ 0) outside the roots:
Answer: x ≤ −23 or x ≥ 4
Scanned GCSE Exam Questions & Worked Solutions

GCSE Past Paper Quadratic inequalities Questions & Answers

Past Paper Example 1 IGCSE Specimen 1H Q17 4 marks
Full Paper (PDF) • Mark Scheme (PDF)
(a) Solve x² + 2x > 6x + 5. (3)
(b) Represent your solution set to part (a) on the number line. (1)
(a) Solve x2 + 2x > 6x + 5 [3 marks]
(b) Represent your solution set to part (a) on the number line below. [1 mark]
–5–4–3–2–10123456x
Worked Solution & Mark Scheme (4 marks)
Part (a) Solution:
x² – 4x – 5 > 0 M1
⇒ (x – 5)(x + 1) > 0 M1
⇒ x < –1 or x > 5 A1
Answer (a): x < -1, x > 5
Part (b) Solution:
Open circles at –1 and 5 with arrows pointing left from –1 and right from 5
–5–4–3–2–10123456x
Answer (b): Open circles at –1 and 5 with arrows pointing left from –1 and right from 5
Past Paper Example 2 AQA Nov 2022 1H Q25 3 marks
Full Paper (PDF) • Mark Scheme (PDF)
(a) Solve x² – 5x – 6 < 0. [2 marks]
25 (b) Show the solution on the number line. [1 mark]
(a) Solve x2 - 5x - 6 < 0 [2 marks]
(b) Show the solution to x2 - 5x - 6 < 0 on the number line. [1 mark]
x–10–9–8–7–6–5–4–3–2–1012345678910
Worked Solution & Mark Scheme (3 marks)
Part (a) Solution:
(x – 6)(x + 1) < 0 M1
⇒ roots at x = –1 and x = 6, so –1 < x < 6 A1
Answer (a): Answer -1 < x < 6
Part (b) Solution:
Open circles at –1 and 6 joined by a line B1
x–10–9–8–7–6–5–4–3–2–1012345678910
Past Paper Example 3 GCSE Jun 2017 3H Q19 3 marks
Full Paper (PDF) • Mark Scheme (PDF)
Solve 2x2 + 3x - 2 > 0
Worked Solution & Mark Scheme (3 marks)
Factorise: (2x – 1)(x + 2) > 0. Critical values are x = –2 and x = 1/2. Since the parabola opens upwards, 2x² + 3x – 2 > 0 for x < –2 or x > 1/2. M2 A1
Answer: x < -2 or x > 12

← 8D Completing the square

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