Description

ALGEBRA – MATHEMATICS F2
Topic 04: Algebra
Form 2 Mathematics Notes | Comprehensive & Exam-Focused

📘 OVERVIEW
This comprehensive set of notes covers Algebra – a core topic in Form 2 Mathematics. It is carefully structured to help students master binary operations, transposition of formulae, quadratic expressions, factorisation, and quadratic equations, with clear worked examples and practice questions for exam preparation.

✨ WHAT THIS DOCUMENT COVERS
1. Introduction to Algebra
Understanding algebra

Real-life applications: budgeting, financial planning, comparing prices, planning journeys, cooking proportions, science, engineering, and technology

2. Binary Operations
Definition: A rule for combining two quantities to produce a unique quantity from the same set

Key points: Symbols used are not standard, may have different meanings in different expressions

Common symbols: Δ, *, ⊙, ●

Basic form: a * b = some rule involving a and b

Worked Examples:

Example 4.1: If a * b = 5a − b, find 6 * 9

Example 4.2: If a * b = a² − b, find y given that 4 * (2 * y) = 4

Example 4.3: If x * y = 4x + 6y, find 6 * 4

Example 4.4: If p * q = p² − q², find 7 * 3

Example 4.5: If a * b = 3a + b, find 5 * 8

Example 4.6: If x * y = x + 3y, find 5 * (8 * 6)

Example 4.7: If p Δ q = 2p + 5q, find 6 Δ 3

Example 4.8: If m Δ n = (m − n)/(m + n), find 3 Δ 2

Binary Operations Practice (Q1–Q10)

3. Transposition of Formulae
Definition: The procedure of expressing a formula in different ways by changing the subject

Common formulae: A = lb, V = πr²h, I = PRT/100, A = ½(a + b)h, T = 2π√(l/g), y = mx + c, s = ut + ½at²

Worked Examples:

Example 4.9: Given y = mx + c, make m the subject

Example 4.10: V = πr²h, make r the subject

Example 4.11: T = 2π√(l/g), make l the subject

Example 4.12: I = PRT/100, make P the subject

Transposition Practice (13 formulae to make given letters the subject)

4. Quadratic Expressions
Definition: An expression whose highest exponent (degree) of the variable is 2

General form: ax² + bx + c, where a ≠ 0

Components: quadratic term, linear term (middle term), constant term

Expansion of linear expressions

Worked Examples:

Example 4.13: Expand (z + 2)(2z − 3)

Example 4.14: Find coefficient of n and n² in (n + 9)(n + 3)

Example 4.15: Expand (6x + 5)²

Important Identities:

(a + b)² = a² + 2ab + b²

(a − b)² = a² − 2ab + b²

a² − b² = (a + b)(a − b)

(x + a)(x + b) = x² + (a + b)x + ab

Special Cases: Linear, quadratic (no middle term), quadratic (no constant)

Expansion Practice (Q1–Q16)

5. Factorisation of Quadratic Expressions
Definition: The process of writing an expression as a product of its factors

5.1 Splitting the Middle Term:

Method: Find two numbers whose sum = b and product = ac; split bx; factor by grouping

Worked Examples:

Example 4.16: Factorise 2x² + 7x + 6

Example 4.17: Factorise 6x² − 11x + 4

Example 4.18: Factorise 2x² + x − 10

Example 4.19: Factorise x² + 6x + 9

Practice Questions (Q1–Q12)

5.2 Difference of Two Squares:

Formula: a² − b² = (a + b)(a − b)

Worked Examples:

Example 4.20: Factorise a² − 25b²

Example 4.21: Factorise x² − 81

Example 4.22: Calculate 50001² − 49999²

Practice Questions (Q15–Q20)

5.3 Perfect Squares:

Formulas: a² + 2ab + b² = (a + b)²; a² − 2ab + b² = (a − b)²

Key property: constant term = (half the coefficient of the middle term)²

Worked Examples:

Example 4.23: Factorise x² − 6x + 9

Example 4.24: Factorise a² + (6/5)a + 9/25

Practice Questions (Q7–Q14)

6. Quadratic Equations
Definition: A quadratic equation is formed when a quadratic expression is equal to a specific value

General form: ax² + bx + c = 0, where a ≠ 0

Roots: The solutions of a quadratic equation

Methods of solving: Factorisation, Completing the square, Quadratic formula

6.1 Solving by Factorisation:

Steps: Write in standard form, factorise, set each factor = 0, solve

Zero Factor Theorem

Worked Examples:

Example 4.25: Solve (x + 4)(x − 3) = 0

Example 4.26: Solve 3x² − x = 0

Example 4.27: Solve 10x² + 9x + 2 = 0

Example 4.28: Solve t² + 6t + 8 = 0

Example 4.29: Solve x² − 9 = 0

Example 4.30: Solve 2h² − 7h = 39

Example 4.33: Rectangular garden problem

Factorisation Practice (Q1–Q3, Q10)

6.2 Solving by Completing the Square:

Method: Move constant, add (half coefficient of x)², write as perfect square, take square root, solve

Worked Examples:

Example 4.34: What must be added to x² + 10x to make a perfect square?

Example 4.35: Solve x² + 2x − 15 = 0

Example 4.36: Solve 2x² − 8x + 3 = 0

Completing the Square Practice

6.3 Solving by Quadratic Formula:

Derivation: x = (−b ± √(b² − 4ac)) / (2a)

Conditions: b² − 4ac > 0 (two distinct real roots), b² − 4ac = 0 (one repeated real root), b² − 4ac < 0 (no real roots) Worked Examples: Example 4.40: Solve 6x² + 11x + 3 = 0 Example 4.41: Solve 5x² − 6x − 1 = 0 Quadratic Formula Practice (Q1–Q18) 7. Chapter Summary Binary operations Quadratic expression Factorisation methods: Splitting middle term, Difference of squares, Perfect squares Quadratic equations: Factorisation, Completing the square, Quadratic formula 8. Revision Exercise 4: Selected Solutions Q1–Q19 with detailed solutions 🎯 WHY THIS RESOURCE IS PERFECT FOR YOU ✅ Well-Organised – Content is broken down into clear, digestible sections ✅ Exam-Focused – Includes worked examples, practice questions, and revision exercises ✅ Comprehensive – Covers binary operations, transposition, quadratic expressions, factorisation, and quadratic equations ✅ Student-Friendly – Simple language with detailed explanations and step-by-step solutions ✅ Revision-Ready – Summary points, formula references, and practice questions ✅ Teacher-Approved – Ideal for classroom use, homework, and exam preparation 📚 WHO IS THIS FOR? Form 2 Students studying Mathematics Teachers preparing lesson notes and assessments Parents supporting their children's learning Anyone interested in algebra and quadratic equations 🛒 GET YOUR COPY TODAY! This is a must-have resource for anyone serious about mastering Form 2 Mathematics. Whether you're revising for exams or teaching in class, these notes will save you time and help you achieve excellent results. MITIHANI POPOTE — Quality educational resources wherever you are.