Study Notes: Quadratic Equations (Class 10)

Overview

Quadratic equations form a critical bridge between linear algebra and higher mathematics in the SOF IMO curriculum. A quadratic equation is any equation of the form ax² + bx + c = 0, where a ≠ 0. This topic appears frequently in both the Mathematical Reasoning section (15–20 marks) and occasionally in the Achievers Section as part of multi-step problems.

Mastery of quadratic equations means knowing **four solution methods**: factorization, completing the square, the quadratic formula, and understanding the discriminant. Each method has its place—factorization is fastest when factors are obvious, completing the square builds conceptual clarity, and the quadratic formula is the universal tool. The discriminant (b² – 4ac) determines the nature of roots without solving, a frequent trap in olympiad questions.

For SOF IMO, expect 2–3 direct questions on solving quadratics, 1–2 on discriminant analysis, and word problems connecting to areas, AP sequences, or geometry. Speed and accuracy in choosing the right method separate high scorers from average performers.

Key Concepts

• **Standard form**: A quadratic equation must be written as ax² + bx + c = 0 (a ≠ 0) before solving. If a = 0, it becomes linear, not quadratic.

• **Roots or zeroes**: The values of x that satisfy the equation. A quadratic can have at most two real roots, though they may be equal or complex.

• **Factorization method**: Express ax² + bx + c as a product of two linear factors. Works cleanly only when roots are rational. Look for two numbers that multiply to ac and add to b.

• **Completing the square**: Transform the equation into (x + p)² = q form. This method reveals the vertex of the parabola and works for all quadratics, though it's algebraically intensive.

• **Quadratic formula**: x = (–b ± √(b² – 4ac)) / 2a. Universal method that works for any quadratic. Memorize this formula—it's your fallback for difficult factorizations.

• **Discriminant Δ**: Defined as Δ = b² – 4ac. It determines root nature: Δ > 0 gives two distinct real roots, Δ = 0 gives two equal real roots (repeated root), Δ < 0 gives no real roots (complex roots, beyond Class 10 scope).

• **Sum and product of roots**: If α and β are roots, then α + β = –b/a and αβ = c/a. Use these relations to form equations from given roots or verify solutions quickly.

• **Word problems translation**: "Consecutive integers", "length exceeds breadth by 5", "sum of a number and its reciprocal" all translate to quadratic setups. After solving, reject negative or non-physical solutions.