ICSE Class 10 Maths: Quadratic Equations — Complete Notes 2026

Standard Form and Roots

• ax² + bx + c = 0 (a ≠ 0); solutions are called roots or zeros
• A quadratic has at most 2 roots; roots satisfy the equation when substituted
• Sum of roots α + β = −b/a; product of roots αβ = c/a

Factorisation Method

• Factorise LHS; if (x−p)(x−q) = 0, then x = p or x = q
• Method: find two numbers whose product = ac and sum = b; split middle term
• Works cleanly for rational roots; discriminant must be a perfect square

Completing the Square

• Transform ax² + bx + c = 0 to (x + p)² = q form
• Step-by-step: divide by a, move c/a to right, add (b/2a)² to both sides, take square root
• Foundation of quadratic formula derivation

Quadratic Formula

• x = (−b ± √(b²−4ac)) / 2a; works for all quadratic equations
• Discriminant D = b²−4ac: D>0 two real roots; D=0 equal roots; D<0 no real roots
• ICSE: always calculate D first before solving

Nature of Roots Using Discriminant

• If D is perfect square: rational roots; if D>0 but not perfect square: irrational roots
• Condition for equal roots: b² = 4ac; find k in equation with equal roots
• Condition for real roots: b² ≥ 4ac

Word Problems — Quadratics

• Area and dimension problems: length is 3 more than breadth; area = 108 cm²
• Time-speed-distance: time for two parts of journey expressed as fractions
• Work problems: A does work in x days; B takes 3 more days; together in 2 days

ICSE Numericals — Quadratics

• Solve 2x² − 5x + 3 = 0 by factorisation
• Find k if 3x² − kx + 4 = 0 has equal roots
• A rectangle has area 48 m²; if length is 2m more than breadth, find dimensions