Algebra: Expanding, Factorising and Equations: Year 8 Study Guide
Year 8 algebra builds the tools for solving problems with more than one step.
To expand an expression, multiply each term inside the brackets by the term outside: 4(2x − 3) = 8x − 12. For two brackets, multiply each term in the first by each term in the second: (x + 2)(x + 5) = x² + 5x + 2x + 10 = x² + 7x + 10. To factorise, take out the highest common factor: 12x + 18 = 6(2x + 3).
A linear equation can be solved by doing the same operation to both sides. When the unknown appears on both sides, collect the variable terms on one side first. Equations model problems, and a formula such as A = ½bh can be rearranged to find any variable.
Expanding and factorising
| Expression | Expanded or factorised | Rule |
|---|---|---|
| 3(x + 4) | 3x + 12 | distribute |
| −2(y − 5) | −2y + 10 | negative times negative |
| (x + 3)(x + 4) | x² + 7x + 12 | multiply each pair |
| 8a + 12 | 4(2a + 3) | highest common factor |
| x² + 6x | x(x + 6) | common factor x |
Solving equations
Solve 3(x − 2) = 2x + 5. Expand: 3x − 6 = 2x + 5. Subtract 2x: x − 6 = 5. Add 6: x = 11. Check: 3 × 9 = 27 and 2 × 11 + 5 = 27. ✓ For fractions, multiply every term by the common denominator: x/3 + 2 = 5 → x + 6 = 15 → x = 9.
Remove them first.
Move them across by inverse operations.
Divide.
Substitute.
Formulas and problems
Rearrange v = u + at for t: t = (v − u) ÷ a. Substitute: if A = ½bh, b = 8 and h = 5, then A = 20. Word problem: Sam is 3 years older than twice Jo's age, and the sum of their ages is 33: 2j + 3 + j = 33 → 3j = 30 → j = 10, so Sam is 23. Define the variable, set up the equation, solve and answer in words.
Worked examples and common mistakes
Example: Factorise 15m − 25: the HCF is 5, so 5(3m − 5). Common mistakes: expanding only the first term, sign errors with negatives and cancelling terms that are added. Check by expanding your factorised answer.
Check your understanding: model answers
1. 10x − 15. 2. 2x = 8, so x = 4. 3. 7(2a + 3).
✨ Key Takeaways
- Distribute to every term.
- Factorise by taking out the HCF.
- Keep equations balanced.
- Check by substitution.
💬 Questions for Discussion & Reflection
- Expand 5(2x − 3).
- Solve 4x + 7 = 2x + 15.
- Factorise 14a + 21.