Why This Matters#
Expansion (multiplying out brackets) is the most used skill in all of high school maths. You will use it in:
- Equations (Grade 10–12)
- Functions and graphs (Grade 10–12)
- Trigonometric identities (Grade 11–12)
- Calculus first principles (Grade 12)
- Finance formulas (Grade 12)
If you can expand quickly and accurately, everything else becomes easier. If you can’t, errors will follow you through every topic.
1. Single Bracket (The Distributive Law)#
Rule: Multiply everything outside by everything inside.
$$ a(b + c) = ab + ac $$Worked Examples#
$3(x + 4) = 3x + 12$
$-2(3x - 5) = -6x + 10$ ← Watch the signs! $(-2) \times (-5) = +10$
$x(x^2 + 3x - 1) = x^3 + 3x^2 - x$
Key point: The negative sign in front of a bracket flips EVERY sign inside: $-(2x - 3) = -2x + 3$
2. Double Brackets (FOIL)#
When multiplying $(x + a)(x + b)$, multiply each term in the first bracket by each term in the second:
- First × First
- Outer × Outer
- Inner × Inner
- Last × Last
Worked Example 1#
$(x + 2)(x + 3)$
$= x^2 + 3x + 2x + 6$
$= x^2 + 5x + 6$
Worked Example 2#
$(2x - 3)(x + 5)$
$= 2x^2 + 10x - 3x - 15$
$= 2x^2 + 7x - 15$
Worked Example 3 (with negatives)#
$(x - 4)(x - 7)$
$= x^2 - 7x - 4x + 28$
$= x^2 - 11x + 28$
Note: $(-4) \times (-7) = +28$. Both negative → positive product.
3. Special Products — Memorise These#
The Square of a Binomial#
$$ (a + b)^2 = a^2 + 2ab + b^2 $$$$ (a - b)^2 = a^2 - 2ab + b^2 $$⚠️ THE #1 EXPANSION ERROR: $(x + 3)^2 \neq x^2 + 9$. You MUST include the middle term $2(x)(3) = 6x$. The correct answer is $x^2 + 6x + 9$.
Worked Examples#
$(x + 5)^2 = x^2 + 10x + 25$
$(3x - 2)^2 = 9x^2 - 12x + 4$ ← Here $a = 3x$ and $b = 2$, so $2ab = 2(3x)(2) = 12x$
$(x - 1)^2 = x^2 - 2x + 1$
The Difference of Squares (Conjugates)#
$$ (a + b)(a - b) = a^2 - b^2 $$The middle terms cancel out because one is $+ab$ and the other is $-ab$.
Worked Examples#
$(x + 4)(x - 4) = x^2 - 16$
$(3x + 2)(3x - 2) = 9x^2 - 4$
$(x + \sqrt{5})(x - \sqrt{5}) = x^2 - 5$
4. Three Terms × Two Terms#
When a trinomial is multiplied by a binomial, distribute each term:
$(x + 2)(x^2 - 3x + 1)$
$= x(x^2 - 3x + 1) + 2(x^2 - 3x + 1)$
$= x^3 - 3x^2 + x + 2x^2 - 6x + 2$
$= x^3 - x^2 - 5x + 2$
5. The Cube of a Binomial (Used in Grade 12 Calculus)#
$$ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 $$You won’t be asked to derive this in Grade 10, but knowing it saves time in Grade 12 first principles.
Shortcut: Use Pascal’s triangle — coefficients are 1, 3, 3, 1.
$(x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3$
🚨 Common Mistakes#
- Forgetting the middle term when squaring: $(x + 3)^2 = x^2 + 6x + 9$, NOT $x^2 + 9$.
- Sign errors with negatives: $(-2)(x - 3) = -2x + 6$, NOT $-2x - 6$. Two negatives make a positive.
- Not collecting like terms: After FOIL, always look for terms you can combine ($3x + 2x = 5x$).
- Distributing only to the first term: $3(x + 4) = 3x + 12$, NOT $3x + 4$.
💡 Pro Tip: The “Check” Method#
After expanding, you can verify your answer by substituting a simple value (like $x = 1$) into both the original and expanded form. They should give the same number.
$(x + 2)(x + 3)$ at $x = 1$: $(3)(4) = 12$ ✓
$x^2 + 5x + 6$ at $x = 1$: $1 + 5 + 6 = 12$ ✓
🔗 Related Grade 10 topics:
- Factorization — the REVERSE of expanding brackets. You need both skills.
- Solving Equations — expanding is often the first step when solving equations with brackets