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Factorise
Standard form: ax²+bx+c=0, a ≠ 0; zero product gives roots
Preserve zero: x²=5x → x(x−5)=0 → 0 or 5; do not divide by x
Interpret
Signs / graphs: x²=49 → ±7; parabola may have 0,1,2 real roots
Context: Width x: x(x+3)=28 → x=4; reject −7 with reason
Precision: Keep enough digits through subsequent work
Complete square
Structure: Higher: x²+6x+5 = (x+3)²−4
Solve: Higher: (x+3)²=4 → x=−1 or −5
Formula
Both roots: Higher: x = (−b ± √(b²−4ac))/(2a); bracket negatives
Discriminant: Higher: Positive: two; zero: repeated; negative: no real root
Choose: Higher: Factorise if possible; square or formula for general case
Connections
Factorise → Interpret: Factored roots must be interpreted in context
Complete square → Formula: Completed squares and discriminants describe real roots
