Quadratics, Simultaneous Equations and Iteration
Solve quadratic and simultaneous systems and use iterative methods with justified accuracy.
How to study for GCSE Mathematics
Use 601 original multiple-choice questions to learn connected methods, explain why they work, practise exact non-calculator execution and use a calculator strategically on designated papers.
Core concepts
Concept 1
Quadratic equations
Exam cue: Choose factorisation, formula or completion of the square for a quadratic; eliminate or substitute for a system; iterate from the stated start value.
Concept 2
Simultaneous equations
Exam cue: Substitute every solution into the original equations and verify that successive iterations support the stated accuracy.
Concept 3
Iteration and numerical solutions
Exam cue: Represent the information clearly, choose a justified method and communicate each step with correct notation and units.
Targeted study blocks
Tier coverage
Foundation core with Higher-tier extensions
Higher includes the quadratic formula, completing the square, simultaneous linear–quadratic systems, iterative approximation and algebraic interpretation of roots.
Calculator structure
Prepare for calculator and non-calculator papers
Any part of a board's specification may be assessed on any paper. Non-calculator does not define a smaller syllabus; questions are designed so exact arithmetic and reasoning are feasible without a calculator.
Risk pitfalls and guardrails
Discarding a valid second root or reporting an iteration without an accuracy check.
Guardrail: Do not hide an invalid model behind arithmetic: check assumptions, signs, bounds, units, scale, required accuracy and the original question.
Giving an unsupported answer when the command requires working, proof, reasoning or interpretation.
Guardrail: Do not hide an invalid model behind arithmetic: check assumptions, signs, bounds, units, scale, required accuracy and the original question.
Assuming a topic belongs only to calculator or non-calculator papers; any specification content can be assessed on any paper.
Guardrail: Any part of a board's specification may be assessed on any paper. Non-calculator does not define a smaller syllabus; questions are designed so exact arithmetic and reasoning are feasible without a calculator.
Memory anchors
Zero-product rule
If ab = 0, then a = 0 or b = 0.
Simultaneous solution
A value pair satisfying both equations.
Iteration check
Successive values must agree sufficiently for the required rounding.
Quadratics, Simultaneous Equations and Iteration: method
Choose factorisation, formula or completion of the square for a quadratic; eliminate or substitute for a system; iterate from the stated start value.
Quadratics, Simultaneous Equations and Iteration: check
Substitute every solution into the original equations and verify that successive iterations support the stated accuracy.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
Shared F/H · AO1 technique · Non-calculator — Solve x²−9x+20=0. What is the larger root?
Shared F/H · AO1 technique · Non-calculator — Solve x²+2x−15=0. What is the smaller root?
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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