Topic explanation
Solving quadratic equations by factorising, completing the square, and using the quadratic formula.
Key formulas & rules
- Factorised form: (x − p)(x − q) = 0 ⇒ x = p or x = q
- Completing the square: x² + bx + c = (x + b/2)² − (b/2)² + c
- Quadratic formula: x = (−b ± √(b² − 4ac)) / (2a) for ax² + bx + c = 0
- Discriminant Δ = b² − 4ac: Δ > 0 two distinct roots; Δ = 0 one repeated root; Δ < 0 no real roots
Common mistakes
- Dividing by x without checking x ≠ 0 — you can lose the root x = 0.
- Sign slips when c is negative in the quadratic formula or when moving terms across the equals sign.
- Giving decimal roots when the question asks for **exact** answers (use surds or fractions).
- Sketching U-shaped parabolas when a < 0 (should be ∩-shaped).
Exam tips
- Always rearrange to ax² + bx + c = 0 before choosing factorising vs formula.
- If factorising fails quickly, use the formula — method marks still flow from correct substitution.
- After solving, substitute back into the original equation if time allows (quick sanity check).
- Link roots to the graph: roots are x-intercepts; turning point relates to completed-square form.
Practice questions
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