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Quad. sim. equations

Free Printable Quad. sim. equations PDF Worksheets, Revision Guide & GCSE Past Paper Questions (Grade 8+)

Download free quad. sim. equations worksheets (PDF), GCSE maths past paper questions and worked solutions for Quad. sim. equations (8A). 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, Maths4Everyone, Corbett Maths, Maths Made Easy, Naiker Maths, 1st Class Maths, Maths Takeaway, Dr Austin, E Classroom and PMT, along with past paper question booklets.

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

Free Quad. sim. equations PDF Worksheets & Answers

Grade 8+ • Topic 8A
Resource Questions Answers
GCSE past paper questions & solutions
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GCSE 9–1 practice booklet & solutions
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MME worksheet & model answers
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GCSE Maths Takeaway exam questions
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PMT (Edxl)
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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 Quad. sim. equations: Explanation, Diagrams & Worked Examples

Download Printable One Page Guide (PDF)

Explanation:

To solve simultaneous equations where one equation is linear and the other is quadratic (such as a circle x2 + y2 = r2 or parabola), always use substitution: rearrange the linear equation to make x or y the subject, then substitute this expression into the quadratic equation.

Expand the brackets carefully, rearrange into standard quadratic form ax2 + bx + c = 0, and solve by factorising or the quadratic formula. Finally, substitute each root back into the linear equation and pair up your (x, y) coordinates clearly.

Example 1: Solve y = x2 − 3x + 4 and y = 2x − 2.

Solution:

Equate y: x2 − 3x + 4 = 2x − 2
Rearrange to 0: x2 − 5x + 6 = 0 ⇒ (x − 2)(x − 3) = 0
So x = 2 or x = 3
In y = 2x − 2: when x = 2, y = 2; when x = 3, y = 4
Answer: (2, 2) and (3, 4)

Example 2: Solve x2 + y2 = 25 and y − 2x = 5.

(0, 5) (−4, −3) x2+y2=25 5

Solution:

Substitute y = 2x + 5 into x2 + y2 = 25:
x2 + (2x + 5)2 = 25 ⇒ 5x2 + 20x = 0
5x(x + 4) = 0 ⇒ x = 0 (y = 5) or x = −4 (y = −3)
Answer: (0, 5) and (−4, −3)
Scanned GCSE Exam Questions & Worked Solutions

GCSE Past Paper Quad. sim. equations Questions & Answers

Past Paper Example 1 GCSE Nov 2023 1H Q21 5 marks
Full Paper (PDF) • Mark Scheme (PDF)
(a) On the grid, draw the graph of x² + y² = 169 (2) (b) Use your graph to find estimates for the solutions of x² + y² = 169 and 2y = 3x (3)
(a) On the grid, draw the graph of x2 + y2 = 169 [2 marks]
–14–14–12–12–10–10–8–8–6–6–4–4–2–222446688101012121414Oyx
(b) Use your graph to find estimates for the solutions of the simultaneous equations
x2 + y2 = 169
2y = 3x [3 marks]
Worked Solution & Mark Scheme (5 marks)
Part (a) Solution:
√169 = 13, draw circle centre (0, 0) radius 13 M1 A1
–14–14–12–12–10–10–8–8–6–6–4–4–2–222446688101012121414Oyx
Answer (a): Circle drawn with centre (0, 0) and radius 13
Part (b) Solution:
Draw line y = 1.5x M2
intersections at (7.2, 10.8) and (–7.2, –10.8) A1
Answer (b): x = 7.2, y = 10.8 and x = -7.2, y = -10.8
Past Paper Example 2 GCSE Nov 2021 1H Q20 3 marks
Full Paper (PDF) • Mark Scheme (PDF)
The diagram shows the graph of x2 + y2 = 30.25
xy–7–7–6–6–5–5–4–4–3–3–2–2–1–111223344556677O
Use the graph to find estimates for the solutions of the simultaneous equations
x2 + y2 = 30.25
y - 2x = 1
Worked Solution & Mark Scheme (3 marks)
Plot line y = 2x + 1 through (0, 1) and (2, 5) M2
Intersections with circle are at (2.1, 5.1) and (–2.9, –4.7). A1
xy–7–7–6–6–5–5–4–4–3–3–2–2–1–111223344556677O
Answer: x = 2.1, y = 5.1 and x = -2.9, y = -4.7
Past Paper Example 3 GCSE Jun 2018 2H Q16 5 marks
Full Paper (PDF) • Mark Scheme (PDF)
(a) Draw graph of x² + y² = 12.25 (2) (b) Hence find estimates for solutions of x² + y² = 12.25 and 2x + y = 1 (3)
(a) On the grid, draw the graph of x2 + y2 = 12.25 [2 marks]
xy–6–6–4–4–2–2224466O
(b) Hence find estimates for the solutions of the simultaneous equations
x2 + y2 = 12.25
2x + y = 1 [3 marks]
Worked Solution & Mark Scheme (5 marks)
Part (a) Solution:
r = √12.25 = 3.5 M1 A1
xy–6–6–4–4–2–2224466O
Answer (a): Circle drawn with centre (0, 0) and radius 3.5
Part (b) Solution:
Draw line y = 1 – 2x M2
intersections with circle are at (2.0, –2.9) and (–1.2, 3.3) A1
Answer (b): x = 2.0, y = -2.9 and x = -1.2, y = 3.3

← 7U Area of a segment

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