Quadratics and Root Conditions
Expose quadratic structure through completing the square, sign diagrams and discriminant conditions.
How to study Higher Mathematics
Practise a skill without a calculator first, connect it to neighbouring techniques, then apply it in unfamiliar contexts with complete working and interpretation.
Core concepts
Concept 1
non-unit leading coefficient
Exam cue: Should the leading coefficient be factored before completing the square?
Concept 2
completed-square form
Exam cue: Where is the parabola above or below the axis?
Concept 3
quadratic inequalities
Exam cue: What discriminant condition matches the stated roots?
Concept 4
discriminant
Concept 5
root nature
Risk pitfalls and guardrails
Completing the square without accounting for the leading coefficient
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Giving roots instead of an inequality interval
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Using b² + 4ac instead of b² − 4ac
Guardrail: Check signs, brackets, domain restrictions, units and whether the answer needs justification.
Memory anchors
Factor the front first
For a non-unit x² coefficient, take it outside the quadratic pair.
Square creates a correction
Add and subtract the same internal square term.
Roots split the sign
Use critical roots and the parabola direction for inequalities.
Discriminant decides
b² − 4ac determines whether roots are two, one or none real.
Tangent means zero
A repeated root corresponds to discriminant zero.
Interval, not endpoints alone
An inequality answer must state the full solution set.
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
Write 2x²+12x+5 in completed-square form.
Write 3x²−18x+11 in completed-square form.
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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