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.

Formulas / Key Facts

1. **Standard quadratic form**: ax² + bx + c = 0, a ≠ 0 2. **Quadratic formula**: x = (–b ± √(b² – 4ac)) / 2a 3. **Discriminant**: Δ = b² – 4ac 4. **Nature of roots**: Δ > 0 → two distinct real roots; Δ = 0 → two equal real roots; Δ < 0 → no real roots 5. **Sum of roots**: α + β = –b/a 6. **Product of roots**: αβ = c/a 7. **Perfect square trinomial**: x² + 2px + p² = (x + p)² 8. **Forming equation from roots**: x² – (sum of roots)x + (product of roots) = 0

Worked Examples

**Example 1: Factorization** Solve: x² – 5x + 6 = 0

*Solution*: Find two numbers that multiply to 6 and add to –5. These are –2 and –3. x² – 5x + 6 = (x – 2)(x – 3) = 0 So x – 2 = 0 or x – 3 = 0 **Roots: x = 2, 3**

**Example 2: Quadratic Formula** Solve: 2x² + 3x – 5 = 0

*Solution*: Here a = 2, b = 3, c = –5. Δ = b² – 4ac = 9 – 4(2)(–5) = 9 + 40 = 49 x = (–3 ± √49) / (2×2) = (–3 ± 7) / 4 x = (–3 + 7)/4 = 1 or x = (–3 – 7)/4 = –10/4 = –5/2 **Roots: x = 1, –5/2**

**Example 3: Discriminant Analysis** For what value of k does kx² + 4x + 1 = 0 have equal roots?

*Solution*: Equal roots means Δ = 0. Δ = b² – 4ac = 16 – 4(k)(1) = 16 – 4k Set 16 – 4k = 0 4k = 16 **k = 4**

**Example 4: Word Problem** The sum of a number and its reciprocal is 13/6. Find the number.

*Solution*: Let the number be x. Then x + 1/x = 13/6. Multiply by 6x: 6x² + 6 = 13x Rearrange: 6x² – 13x + 6 = 0 Factor: (2x – 3)(3x – 2) = 0 x = 3/2 or x = 2/3 **Both values are valid (reciprocals of each other)**

Common Mistakes

• **Forgetting a ≠ 0**: Writing 0x² + 3x + 5 = 0 is linear, not quadratic. Always check the coefficient of x² before applying quadratic methods.

• **Sign errors in the quadratic formula**: The formula is –b, not +b. Many students write x = (b ± √Δ)/2a and lose both roots. Write the negative sign explicitly.

• **Incorrect discriminant interpretation**: Δ = 0 means **two equal roots**, not one root. The root is repeated. Also, Δ < 0 means no real solution, but some students attempt to simplify √(–4) within real numbers.

• **Rejecting valid negative roots in word problems**: Not all word problems require positive answers. Read the context—consecutive integers, temperatures, and debts can be negative. Only reject physically impossible values (negative lengths/ages).

• **Incomplete factorization**: After factoring to (2x – 3)(x + 1) = 0, students write x = 2x – 3, forgetting to set each factor to zero separately. Always split: 2x – 3 = 0 **and** x + 1 = 0.

Quick Reference

• **Quadratic form**: ax² + bx + c = 0; ensure a ≠ 0
• **Factorization**: Works when roots are rational; find factors of ac that add to b
• **Formula**: x = (–b ± √(b² – 4ac)) / 2a — universal method
• **Discriminant**: Δ = b² – 4ac tells root nature before solving
• **Sum/Product of roots**: –b/a and c/a respectively — use to verify or form equations
• **Always rearrange to standard form** before applying any method — move all terms to one side