Quadratics, Polynomials and Inequalities
Solving and interpreting quadratic and polynomial equations, simultaneous systems and linear, quadratic or rational inequalities.
How to study A-level Mathematics
Define the objects and conditions, select a representation, execute a justified method, then validate and interpret the result.
Core concepts
Concept 1
Completing the square and the discriminant connect algebraic form to roots, extrema and graphs.
Exam cue: Choose factorisation, completing the square or the formula according to the required form.
Concept 2
Polynomial roots and factors support exact solution, factorisation and the construction of related equations.
Exam cue: Substitute a reduced expression back into simultaneous equations and test every candidate.
Concept 3
An inequality solution is a set of intervals determined by critical values and sign, not merely a list of boundary points.
Exam cue: Use a sign diagram or graph to select the correct inequality intervals.
Risk pitfalls and guardrails
Using the discriminant without linking its sign to the number of real roots.
Guardrail: Do not replace proof with examples, model conditions with unstated assumptions, or mathematical interpretation with unverified calculator output.
Keeping extraneous roots introduced by squaring or substitution.
Guardrail: Do not replace proof with examples, model conditions with unstated assumptions, or mathematical interpretation with unverified calculator output.
Reversing or failing to reverse an inequality when multiplying by a negative quantity.
Guardrail: Do not replace proof with examples, model conditions with unstated assumptions, or mathematical interpretation with unverified calculator output.
Memory anchors
Discriminant
For ax²+bx+c the discriminant b²-4ac determines the real-root pattern.
Repeated Root
A repeated real root occurs when the discriminant equals zero.
Simultaneous System
A simultaneous solution satisfies every equation in the system.
Critical Value
A critical value is where an inequality expression is zero or undefined.
Sign Diagram
A sign diagram records the sign of an expression between its critical values.
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
Solve x²−7x+12=0.
Find the discriminant of 3x²−4x+5.
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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