Expanding Brackets in Algebra
Expanding an expression means removing brackets by multiplying each term inside the bracket by the term outside. It is the reverse process of factorising.
Single Bracket Expansion
To expand a(b + c), multiply a by every term inside the bracket: a times b + a times c. Every term inside gets multiplied — no skipping!
3 times x + 3 times 4 = 3x + 12.
-2 times 3a = -6a. -2 times -5 = +10. Answer: -6a + 10.
4x times 2x = 8x squared. 4x times -3y = -12xy. 4x times 1 = 4x. Answer: 8x squared - 12xy + 4x.
Double Bracket Expansion (FOIL)
To expand (a + b)(c + d), multiply each term in the first bracket by each term in the second bracket, then collect like terms. The FOIL method helps: First, Outside, Inside, Last.
First: x times x = x squared. Outside: x times 2 = 2x. Inside: 3 times x = 3x. Last: 3 times 2 = 6. Total: x squared + 2x + 3x + 6 = x squared + 5x + 6.
2x times x = 2x squared. 2x times 4 = 8x. -1 times x = -x. -1 times 4 = -4. Total: 2x squared + 8x - x - 4 = 2x squared + 7x - 4.
Special Products
| Pattern | Result | Example |
|---|---|---|
| (a + b) squared | a squared + 2ab + b squared | (x+3) squared = x squared + 6x + 9 |
| (a - b) squared | a squared - 2ab + b squared | (x-4) squared = x squared - 8x + 16 |
| (a + b)(a - b) | a squared - b squared | (x+5)(x-5) = x squared - 25 |
Common Mistakes
- Forgetting to multiply the term outside by every term inside: 3(x + 4) does not equal 3x + 4.
- Sign errors when a negative is outside: -2(x - 3) = -2x + 6, not -2x - 6.
- Skipping the like-terms step after FOIL and leaving four terms instead of three.
Key Takeaways
- Every term inside the bracket must be multiplied by the term outside.
- For double brackets, use FOIL: First, Outside, Inside, Last, then collect like terms.
- Learn the three special patterns — they appear constantly in later algebra.
Practice Questions
- Expand 5(2y - 3).
- Expand -3x(x + 4).
- Expand (x + 6)(x - 2).
- Expand (2a + 3)(a - 5).
- Expand and simplify (x + 4) squared - (x - 1) squared.