Ch 1 · 2.1–2.2 Introduction to Algebra and Algebraic Manipulation · IGCSE Mathematics 0580
Topic 1 of 9 in Algebra and Graphs — IGCSE Mathematics 0580 — 44 lessons.
Introduction
Algebra lets one statement cover every case: "the perimeter of a rectangle is 2l + 2w" works for any rectangle. This lesson covers substitution, expanding brackets and the reverse process, factorising, which is the key to solving equations and simplifying fractions later. (Syllabus sections 2.1 and 2.2.)
Key ideas — Substitution
Replace each letter by its value, using brackets around negatives: with a = 3, b = -2, 2a^2 - 5b = 2(9) - 5(-2) = 28. Square only the letter, not the coefficient: 2a^2 = 2 × 9.
🧑🏫 Teacher's explanation
Think of algebra like a recipe where letters are just placeholders for ingredients. If a recipe calls for 'x' amount of flour and 'y' amount of sugar, and you decide that 'x = 5' and 'y = 2', you simply swap the letters for those numbers to find the total. The letters don't change the structure of the math; they just wait for you to give them a value.
📖 Definitions
🔍 Worked example — Easy
Evaluate 5x - 3y when x = 4 and y = 2.
🔍 Worked example — Standard
Evaluate 3a^2 - 4b when a = -3 and b = -2.
🔍 Worked example — Exam-style
A formula for the area of a shape is A = 1/2bh. Calculate A when b = 10 and h = 4.5.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Simplifying and expanding
Collect like terms (same letters and powers): 2a^2 + 3ab - 1 + 5a^2 - 9ab + 4 = 7a^2 - 6ab + 3. Expand by multiplying every term in one bracket by every term in the other:
🧑🏫 Teacher's explanation
Think of algebra as a way to describe patterns without needing specific numbers. Instead of saying "two apples and three apples," we say 2a + 3a = 5a. The letters are just placeholders for numbers we don't know yet.
📖 Definitions
🔍 Worked example — Standard
Expand and simplify: 4(3x - 5) - 2(x + 4)
🔍 Worked example — Exam-style
Expand (x - 3)(2x + 5)(x - 1)
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Factorising
Factorise fully means no common factor is left.
🧑🏫 Teacher's explanation
Factorising is essentially the reverse of expanding brackets. If expanding is like multiplying two numbers together to get a single product, factorising is like taking that product and breaking it back down into its original building blocks. Think of it like taking a Lego castle and breaking it down into the individual bricks that were used to build it.
📖 Definitions
eq 0$. |
🔍 Worked example — Easy
Factorise fully: 12x^2y - 18xy^2
🔍 Worked example — Standard
Factorise fully: x^2 - 49
🔍 Worked example — Exam-style
Factorise fully: 2x^2 + 7x + 3
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Completing the square
x^2 + bx + c = (x + b/2 )^2 - (b/2 )^2 + c
🧑🏫 Teacher's explanation
Completing the square is a way of rewriting a quadratic expression so that it is written as a single squared bracket plus or minus a constant. Think of it like rearranging a messy room: instead of having items scattered everywhere (the ax^2 + bx + c form), you are packing them into one organized box (the (x+p)^2 + q form). This makes it much easier to see the 'turning point' or the minimum/maximum value of the expression.
📖 Definitions
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples
(2x - 3)^2 - (x + 1)(x - 4) = 4x^2 - 12x + 9 - (x^2 - 3x - 4) = 3x^2 - 9x + 13.
🔍 Worked example — Easy
Expand and simplify: (3x - 2)^2 - (x + 4)(x - 1)
🔍 Worked example — Standard
Factorise fully: 27x^2 - 75
🔍 Worked example — Exam-style
Express x^2 - 8x + 12 in the form (x + p)^2 + q
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 16 + 24 = 40. (2) 2x^2 + 7x - 15. (3) 3a(x - 2y) + b(x - 2y) = (3a + b)(x - 2y). (4) (5x + 1)(2x - 3). (5) (x - 4)^2 - 13.