Linear Equations, Algebraic Identities and Factorisation: Class 8 Study Guide
Algebra turns a problem in words into an equation we can solve. Identities and factorisation make that faster.
A linear equation in one variable has only one variable, with the highest power 1, such as 3x + 5 = 20. We solve it by performing the same operation on both sides. Variables can be on both sides: 5x − 3 = 2x + 9 → 3x = 12 → x = 4. Equations with fractions can be cleared by multiplying by the LCM of the denominators.
An identity is true for all values of the variable. The standard ones are: (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b²; (a + b)(a − b) = a² − b². Factorisation writes an expression as a product of factors; it is the reverse of multiplying out.
Standard identities
| Identity | Example |
|---|---|
| (a + b)² = a² + 2ab + b² | (x + 3)² = x² + 6x + 9 |
| (a − b)² = a² − 2ab + b² | (2y − 1)² = 4y² − 4y + 1 |
| (a + b)(a − b) = a² − b² | (x + 5)(x − 5) = x² − 25 |
| (x + a)(x + b) = x² + (a + b)x + ab | (x + 2)(x + 3) = x² + 5x + 6 |
Factorising
First take out any common factor: 6x² + 9x = 3x(2x + 3). Then look for an identity: x² − 49 = (x + 7)(x − 7), and x² + 10x + 25 = (x + 5)². For x² + 7x + 12, find two numbers that multiply to 12 and add to 7: 3 and 4, so (x + 3)(x + 4). Check by multiplying out.
Take it out.
Two terms? Try a² − b².
Try a perfect square or two numbers.
Multiply the factors back.
Word problems
Write an equation from the story. Example: The sum of two consecutive numbers is 91: x + (x + 1) = 91 → x = 45, so the numbers are 45 and 46. Example: A father's age is 3 times his son's. In 12 years, he will be twice as old: 3s + 12 = 2(s + 12) → s = 12. Always check that the answer fits the story.
Worked examples and common mistakes
Example: Evaluate 103² with an identity: (100 + 3)² = 10000 + 600 + 9 = 10609. Common mistakes: writing (a + b)² as a² + b² (the middle term is missing), sign errors when moving terms and forgetting a common factor. Test with a number such as a = 2, b = 1.
Check your understanding: model answers
1. 4x = 16, so x = 4. 2. x² + 8x + 16. 3. (x + 6)(x − 6).
✨ Key Takeaways
- Do the same to both sides.
- (a + b)² has a middle term 2ab.
- a² − b² = (a + b)(a − b).
- Factorising is the reverse of expanding.
💬 Questions for Discussion & Reflection
- Solve 7x − 4 = 3x + 12.
- Expand (x + 4)².
- Factorise x² − 36.