CBSE Class 10 Maths: Quadratic Equations — Complete Notes 2026

Standard Form and Roots

• Standard form: ax² + bx + c = 0 where a ≠ 0; a, b, c are real coefficients
• Roots: values of x satisfying the equation; a quadratic has exactly 2 roots (real or complex)
• Verification: substitute roots back into equation; both must satisfy

Factorisation Method

• Split the middle term: find two numbers with product = ac and sum = b; rewrite and factor
• Example: 2x² + 7x + 3 = 0 → 2x² + 6x + x + 3 → 2x(x+3) + 1(x+3) = 0 → roots: −3 and −½
• Always check: a×c = 6; 6 = 6×1; 6+1 = 7 ✓

Completing the Square

• Make coefficient of x² = 1 (divide by a); move constant to RHS; add (b/2a)² to both sides
• Example: x² + 6x − 7 = 0 → (x+3)² = 16 → x = −3 ± 4 → x = 1 or −7
• Foundation for deriving the quadratic formula

Quadratic Formula

• x = [−b ± √(b²−4ac)] / 2a; works for ALL quadratic equations
• Discriminant D = b² − 4ac: D > 0 → 2 distinct real roots; D = 0 → 2 equal roots; D < 0 → no real roots
• CBSE: always state discriminant value and nature of roots even if formula used

Nature of Roots

• D > 0 and perfect square → rational roots; D > 0 not perfect square → irrational roots
• D = 0: equal roots, each root = −b/2a; common in CBSE questions asking 'find k for equal roots'
• Sum of roots: α + β = −b/a; Product of roots: αβ = c/a — used to form equations and check roots

Word Problems (CBSE Type)

• Consecutive integers: n(n+1) = 72 → n² + n − 72 = 0 → n = 8 (consecutive: 8 and 9)
• Speed/time/distance: train problems — (d/t − d/(t+x) = 1) form quadratic
• Area problems: rectangle with given perimeter and area → quadratic in one variable

CBSE Board Exam Tips

• Quadratic Equations is 4–6 marks in CBSE Board; discriminant and word problems most frequently tested
• Show all steps for factorisation; state roots clearly at end: 'x = 3 or x = −2'
• For no real roots: state D < 0 and write 'equation has no real roots' — do not attempt further