Expanding means removing brackets by multiplying out. It’s one of the most used skills in engineering algebra — every formula manipulation, every quadratic equation, every area calculation uses it.
Single bracketsDouble brackets (FOIL)Grid methodPerfect squaresDifference of two squares
What is expanding?Removing brackets by multiplication
A bracket means “everything inside is multiplied by what’s outside”. Expanding means doing that multiplication to remove the bracket.
a(b + c) = ab + ac
(a + b)(c + d) = ac + ad + bc + bd
💡 Think of the distributive law: sharing the multiplication across every term inside. Like distributing sweets equally to everyone in a group!
📌 The result has the same value as the original — expanding just writes it in a different form (without brackets).
Why do engineers expand brackets?
Solve equations — you must expand before you can collect like terms and solve
Work with formulae — F = ma(1 + kt) needs expanding before substituting numbers
Find areas & volumes — (L + 2)(W + 2) gives the expanded area of a bordered shape
Prove relationships — showing two expressions are equal requires expanding both
⚠️ You cannot just “drop the brackets” — 3(2x + 5) ≠ 3 + 2x + 5. The 3 multiplies EVERY term inside!
1️⃣
Single Bracket
The simplest type — one term multiplied by everything inside a single bracket. Multiply each term inside by the term outside, one by one.
Rule: a(b + c) = ab + ac Every term inside gets multiplied by the term outside.
Step-by-step method
Identify the term outside the bracket
Multiply it by the first term inside — write the result
Multiply it by the second term inside — write the result with the correct sign
Continue for any further terms inside
Simplify by collecting like terms if possible
✅ Always carry the sign of each term — if a term inside is negative, the result of multiplying it will be negative.
Worked Examples
Expand: 5(3x − 4)
5 × 3x: = 15x
5 × (−4): = −20
Answer:15x − 20 ✅
Expand: x(x + 7)
x × x: = x² (x×x = x²)
x × 7: = 7x
Answer:x² + 7x ✅
Expand and simplify: 3(2x + 1) + 2(x − 5)
First: 3(2x+1) = 6x + 3
Second: 2(x−5) = 2x − 10
Collect: 6x + 3 + 2x − 10
Answer:8x − 7 ✅
⚠️ Most common mistake: Only multiplying the first term inside.
3(2x + 5) ≠ 6x + 5 ❌ (forgot to multiply the 5!)
3(2x + 5) = 6x + 15 ✅ (both terms multiplied)
2️⃣
Double Brackets
When two brackets are multiplied together, every term in the first bracket must multiply every term in the second. With two terms in each bracket, that gives us 4 multiplications.
Rule: (a + b)(c + d) = ac + ad + bc + bd Every term from the first × every term from the second.
F = First O = Outer I = Inner L = Last Then collect: x² + 4x + 3x + 12 = x² + 7x + 12
Step-by-stepEvery multiplication listed clearly
Multiply the first terms together
Multiply the outer terms (first of bracket 1 × last of bracket 2)
Multiply the inner terms (last of bracket 1 × first of bracket 2)
Multiply the last terms together
Collect like terms — the two middle terms usually combine
✅ The two middle terms (Outer and Inner) are always like terms and can be added together.
Worked Examples
Expand: (x + 5)(x + 2)
First: x × x = x²
Outer: x × 2 = 2x
Inner: 5 × x = 5x
Last: 5 × 2 = 10
Collect:x² + 7x + 10 ✅
Expand: (2x + 3)(x + 4)
First: 2x × x = 2x²
Outer: 2x × 4 = 8x
Inner: 3 × x = 3x
Last: 3 × 4 = 12
Collect:2x² + 11x + 12 ✅
📐
The FOIL Method
FOIL is a memorable acronym for the order to do the 4 multiplications when expanding double brackets: First, Outer, Inner, Last.
Try it yourselfLive FOIL calculator
ℹ️ How to use: Type numbers into the four boxes to set up your double brackets (ax + b)(cx + d). The FOIL expansion and simplified answer update as you type.
(
coeff. of x
x +
constant
) (
coeff. of x
x +
constant
)
👆This builds the bracket pair (coeff×x + constant)(coeff×x + constant). Change any of the four boxes — the F/O/I/L steps and final answer update instantly. Leave "coeff. of x" as 1 for a plain x, or try 2, 3 etc. Try a −5 constant to see how negatives carry through.
FOIL ChecklistNever miss a multiplication
F → first term in ( ) 1 × first term in ( ) 2
O → first term in ( ) 1 × last term in ( ) 2
I → last term in ( ) 1 × first term in ( ) 2
L → last term in ( ) 1 × last term in ( ) 2
💡 After FOIL you always have 4 terms. Then collect like terms (usually the O and I terms combine) to get 3 terms.
Worked Examples
Expand: (x − 5)(x + 2)
F: x × x = x²
O: x × 2 = 2x
I: (−5) × x = −5x
L: (−5) × 2 = −10
Collect:x² − 3x − 10 ✅
Expand: (3x − 2)(2x + 5)
F: 3x × 2x = 6x²
O: 3x × 5 = 15x
I: (−2) × 2x = −4x
L: (−2) × 5 = −10
Collect:6x² + 11x − 10 ✅
⋟
The Grid Method
The grid (or box) method is a visual alternative to FOIL. It’s brilliant for larger brackets and for students who prefer to see every multiplication laid out clearly. Every cell in the grid is one multiplication.
How to draw the grid
Draw a 2×2 table (or larger for bigger brackets)
Write the terms of the first bracket as column headers
Write the terms of the second bracket as row headers
Fill each cell by multiplying its row header × column header
Add all cells together and collect like terms
💡 The grid method works for any size brackets — 3 terms in each? Use a 3×3 grid. It’s impossible to miss a term!
Worked Example(2x + 1)(3x + 5) using the grid
★
Special Cases
Two patterns appear so often in engineering algebra that they’re worth memorising. They save time and are tested directly in T Level exams.
Perfect Square(a + b)² = a² + 2ab + b²
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b² (middle term negative)
Expand (x + 4)²
a=x, b=4: a² + 2ab + b²
a²: x²
2ab: 2 × x × 4 = 8x
b²: 4² = 16
Answer:x² + 8x + 16 ✅
⚠️ Very common error: (x + 4)² ≠ x² + 16 ❌ The middle term 2ab = 8x is always there — never forget it!
Difference of Two Squares(a + b)(a − b) = a² − b²
(a + b)(a − b) = a² − b²
The middle terms cancel (+ab and −ab = 0). Only the squares remain!
Expand (3x + 2)(3x − 2)
Pattern: a=3x, b=2 → a² − b²
a²: (3x)² = 9x²
b²: 2² = 4
Answer:9x² − 4 ✅
−
Negatives & Mixed Signs
Signs are where most students lose marks. A careful, consistent approach to negative terms prevents almost all errors.
The Sign RulesAlways apply these when multiplying
💡 Same signs → positive. Different signs → negative. This is the single most important rule to remember when expanding!
Negative outside the bracket
⚠️ A negative sign outside flips ALL signs inside!
−3(2x − 5)
= (−3)×2x + (−3)×(−5)
= −6x + 15
Expand −4(x − 3)
(−4)×x: = −4x
(−4)×(−3): = +12 (neg×neg=pos)
Answer:−4x + 12 ✅
Mixed signs in double brackets
Expand (x − 3)(x − 5)
F: x × x = x²
O: x × (−5) = −5x
I: (−3) × x = −3x
L: (−3)×(−5) = +15 ✔
Collect:x² − 8x + 15 ✅
Expand (2x − 3)²
Pattern: (a−b)² = a²−2ab+b²
a²: (2x)² = 4x²
2ab: 2×2x×3 = 12x
b²: 3² = 9
Answer:4x² − 12x + 9 ✅
⚙️
Applications
Where this topic is used in engineering, manufacturing, maintenance and daily life.
⚡Electrical:P=(I+2)² = I²+4I+4. Expand before differentiating.
🏗️Construction:Floor area (x+3)(x+4) — expand for L-shaped room.
🔧Mechanical:Bending moment M=F(L−x). Expand to integrate.
📊Business:Revenue=(p−5)(200−2p). Expand to maximise.
🔨 Beam & Frame Dimensions
A rectangular steel frame has inner dimensions (x + 4) by (x + 3) metres. Find the area expression.
Test your knowledge — 10 questions, instant feedback.
⚡ Expanding Blitz
ℹ️ How to play: A question appears on screen. Click the answer you think is correct from the four options. Your score and streak update after each question. Click Next to move on.
Score: 0Streak: 0 🔥Q 1/10
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👆Click an answer button to lock it in — correct answers turn green, wrong ones turn red and reveal the right answer · Build a streak 🔥 by answering correctly in a row · Press Play Again after question 10 to reshuffle and try for a higher score.
Score: 0/10
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Practice Questions
15 questions across 3 tiers. Try each one before revealing the answer — especially the sign rules!
🟢 Tier 1 — Foundations
ℹ️ How to use: Work through each question and write down your answer. When you are ready, click Show Answer to reveal the full worked solution and mark scheme.