Solving Quadratic Equations
Select factorisation, graph or the supplied quadratic formula and interpret every valid solution.
How to study National 5 Mathematics
Secure one operational skill at a time, connect it to neighbouring techniques, then apply it in unfamiliar contexts with complete working and a checked conclusion.
Core concepts
Concept 1
A quadratic must be written equal to zero before factorised-product reasoning or the quadratic formula is applied.
Exam cue: Rearrange into ax² + bx + c = 0 and record coefficients with their signs.
Concept 2
Factorisation uses the zero-product rule; a graph solution reads the x-coordinates where y = 0.
Exam cue: Use both plus and minus branches of the formula.
Concept 3
The formula x = (−b ± √(b² − 4ac))/(2a) is supplied on both papers and requires correct identification of a, b and c.
Exam cue: Reject a mathematical root only when the context makes it invalid, and state why.
Risk pitfalls and guardrails
Using coefficients before collecting all terms on one side.
Guardrail: Check signs, brackets, restrictions, units, accuracy and whether the answer needs a reason.
Forgetting one factorised root or one ± branch.
Guardrail: Check signs, coefficients and both sides of every algebraic transformation.
Dropping the entire denominator 2a from part of the numerator.
Guardrail: Check signs, brackets, restrictions, units, accuracy and whether the answer needs a reason.
Memory anchors
Quadratic start
Write the equation as ax² + bx + c = 0.
Zero-product rule
If pq = 0, then p = 0 or q = 0.
Graphical root
An x-coordinate where the graph meets the x-axis.
Quadratic formula
x = (−b ± √(b² − 4ac))/(2a).
Coefficient signs
Record a, b and c exactly from the zero form.
Context filter
Calculate all roots, then justify any rejected value.
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 − 5x + 6 = 0.
Solve x² + x − 12 = 0.
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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