Practice Questions: Expanding and Factorising Algebra
How to use this quiz: Open one topic at a time, type your answer, then check your working. Each topic has 10 questions with answers and explanations.
1. Expanding single brackets
2. Expanding and collecting like terms
3. Expanding double brackets
4. Factorising by common factor
5. Factorising quadratic expressions
6. Factorising harder quadratics
7. Difference of two squares
8. Mixed expansion and factorising challenge
Quick Summary for Expanding and Factorising algebraic expressions
Expanding removes brackets. Factorising puts expressions back into brackets.
Students usually learn basic expansion first, then common-factor factorising,
double brackets, quadratic factorising and difference of two squares.
| Skill | What it means | Example |
|---|---|---|
| Expanding | Remove brackets | 3(x + 4) = 3x + 12 |
| Factorising | Put into brackets | 3x + 12 = 3(x + 4) |
| FOIL | Expand two brackets | (x + 2)(x + 3) = x² + 5x + 6 |
| DOTS | Difference of two squares | x² – 25 = (x + 5)(x – 5) |
How Expansion and Factorisation Progress in the NESA Syllabus
Expansion and factorisation, along with basic algebra and substitution, are essential pillars of the subject. They form the foundation for mastering quadratic equations and parabolas within the NESA curriculum. Because the syllabus is structured around multi-year Stages rather than rigid single years, individual schools have the flexibility to set their own teaching timelines (Scope and Sequence).
However, a typical progression across most schools follows this framework:
Stage 4 (Years 7–8): Students are introduced to algebraic expansion (distributive law) and basic factorisation, such as finding the highest common factor.
Stage 5 (Years 9–10): Students move into advanced applications, including the expansion and factorisation of monic and non-monic quadratic expressions, as well as the Difference of Two Squares (DOTS).
Note: Exact timing varies by school. Some schools may introduce these concepts earlier or later within their Stage 4 and Stage 5 programs.
In this guide, we will break down the essential methods, from expanding single brackets to conquering harder quadratics, giving you the tools you need to succeed in equations, graphing, and beyond.
1. Expanding Single Brackets
Expanding a single bracket involves multiplying the term outside the bracket by every term inside the bracket. This relies on the distributive law.
Example: 8(x + 5)
The 8 must multiply both terms inside the bracket:
8 times x = 8x
8 times 5 = 40
Result: 8(x + 5) = 8x + 40
Common Mistake: Many students multiply the first term but forget the second. Always remember to distribute the outside term to everything inside the bracket!
2. Expanding and Collecting Like Terms
After expanding brackets, simplify the expression by adding like terms. Like terms are terms that share the exact same variable part (e.g., 2x and 5x are like terms, but x and x² are not).
Example: 4(x + 3) + 6x
Step 1: Expand the brackets:
4(x + 3) + 6x => 4x + 12 + 6x [Since, 4(x + 3) = 4x + 12]
Step 2: Add like terms: 4x + 12 + 6x = 10x + 12 [Since, 4x+6x = 10x]
Final Answer: 10x + 12
Simplifying your final expression is critical for scoring full marks in NESA algebra exams.
3. Expanding Double Brackets (The FOIL Method)
A popular way used by NSW schools to expand double brackets is the FOIL method: First, Outer, Inner, Last. In this, Multiply every term in the first bracket by every term in the second bracket.
Example: (x + 2)(x + 3)
Multiply each term carefully:
First: x times x = x²
Outer: x times 3 = 3x
Inner: 2 times x = 2x
Last: 2 times 3 = 6
Combine the results: x² + 3x + 2x + 6
Now, add the like terms (3x and 2x):
Final Answer: x² + 5x + 6
4. Factorising by a Common Factor
Factorising is the exact opposite of expanding. When factorising by a common factor, we look for the highest common factor (HCF) that divides evenly into all parts of the expression.
Example 1: 5x + 15
Both terms can be divided by 5. We bring the 5 outside the bracket: 5(x + 3)
(You can verify this by expanding it back out!)
Example 2: 8x² + 4x
Both terms share a common factor of 4x. Hence, 4x is brought outside the bracket: 4x(2x + 1)
We can double-check by expanding the answer which should give back the original expression.
5. Factorising Simple Quadratic Expressions
A standard quadratic expression is in the form of ax² + bx + c.
To factorise a simple (monic) quadratic, you need to find two numbers that:
Multiply to give the last number (c)
Add to give the middle number (b)
Example: x² + 5x + 6
We need two numbers that multiply to 6 and add to 5. Those numbers are 2 and 3.
Result: (x + 2)(x + 3)
6. Factorising Harder (Non-Monic) Quadratics
Harder quadratics have a coefficient greater than 1 in front of the x² (e.g., ax² + bx + c).
To factorise a simple (non-monic) quadratic, you need to find two numbers that:
Multiply to give the last number (ac)
Add to give the middle number (b)
Example: 2x² + 7x + 3
We need two numbers that multiply to 6 and add to 7. Those numbers are 6 and 1.
Hence, 2x² + 7x + 3
=> 2x² + 6x + x + 3 (We factorize 2x² + 6x and x+3 separately)
=> 2x(x+3) + 1(x+3) (Since x+3 is the common term, we remove it outside the bracket)
Result is: (2x + 1)(x + 3)
7. The Difference of Two Squares (DOTS)
The difference of two squares is a special factorisation rule that NSW math students must memorize. It applies when you have two squared terms separated by a minus sign: a² – b².
The Rule: a² – b² = (a + b)(a – b)
Example 1: x² – 25
Since 25 is 5², the expression becomes: x² – 5²
This fits the rule perfectly for a² – b² where a is x and b is 5
Example 2: 4x² – 9
Here, 4x² – 9 can be written as (2x)² – 3²
Note: This rule only works when there is a subtraction sign. You cannot factorise the sum of two squares with real numbers!
8. Mixed Expansion and Factorising Challenge
In exam settings, questions won’t always be grouped by type. You need to be able to identify the correct mathematical operation based on the terminology.
If the question says “Expand”: Your job is to remove the brackets.
Question: Expand 4(x + 2)
Answer: 4x + 8
If the question says “Factorise”: Your job is to put the expression back into brackets.
Question: Factorise 4x + 8
Answer: 4(x + 2)
Frequently Asked Questions about Expanding and Factorising algebraic expressions
What expansion and factorisation methods do NSW Year 9 and Year 10 students need to know?
Year 9 students usually need to understand how to expand brackets and factorise simple algebraic expressions.
For expansion, students learn how to multiply a term outside the bracket with every term inside the bracket. For example:
3(x + 4) = 3x + 12
For factorisation, students learn how to find the common factor and place it outside the bracket. For example:
6x + 12 = 6(x + 2)
Some schools may also introduce more advanced algebra, such as expanding double brackets and factorising simple quadratic expressions. However, many students are expected to become more confident with quadratic expansion and factorisation in Year 10.
Why is factorising quadratics so important in the NESA Stage 5 syllabus?
Factorising quadratics is important because quadratic equations are a major topic in Stage 5 Mathematics. Students often need factorisation to solve quadratic equations, simplify expressions, and understand the connection between algebra and graphs.
Quadratics are also closely connected to parabolas. When students study parabolas, they need to understand how different forms of a quadratic expression show different information about the graph.
For example:
y = x² + 5x + 6
can be factorised as:
y = (x + 2)(x + 3)
This helps students find the x-intercepts of the parabola. That is why expanding and factorising are not just isolated algebra skills. They are foundation skills for Year 10 algebra, quadratic equations and parabola graphing.
How can a private maths tutor help my child with NSW Year 9 and Year 10 algebra, expansion and factorisation?
A private maths tutor can help by first checking whether the student understands the basics of algebra, brackets, multiplication and common factors.
A good tutor will not just give random questions. They will explain the full structure of the topic first, so the student understands what expansion and factorisation actually mean. Then they will teach each method step by step, including expanding single brackets, collecting like terms, factorising common terms and moving towards quadratic expressions when the student is ready.
After that, the student needs regular practice. Expansion and factorisation become easier when students see many examples and learn how to recognise the method needed for each question.
At Aussie Math Tutor NSW, we focus on clear explanations, step-by-step working and repeated practice so students can become more confident with algebra.
Do your practice tests cover the exact NSW syllabus?
Our practice questions are created to support the NSW Mathematics syllabus and are based on common school-style questions, past exam patterns and the skills students are expected to develop in Years 9 and 10.
The practice tests are designed to help students revise important skills such as expanding brackets, factorising common terms, expanding double brackets and factorising quadratic expressions.
However, schools may teach topics in a slightly different order, and the syllabus or assessment style may change over time. That is why students should also follow their school teacher’s guidance and use our practice tests as extra support for revision and confidence-building.
What is the difference between expanding and factorising?
Expanding means removing brackets by multiplying the term outside the bracket with each term inside the bracket.
Example:
4(x + 3) = 4x + 12
Factorising is the reverse process. It means taking out the common factor and putting the expression back into brackets.
Example:
4x + 12 = 4(x + 3)
A simple way to remember it is:
Expanding opens brackets. Factorising creates brackets.
Why do students find expanding and factorising difficult?
Many students struggle with expanding and factorising because they try to memorise steps without understanding what is happening.
Common mistakes include:
- multiplying only the first term inside the bracket
- forgetting negative signs
- not collecting like terms properly
- not recognising the highest common factor
- confusing expansion with factorisation
Once students understand that expanding and factorising are opposite processes, the topic becomes much easier. Regular practice also helps students recognise patterns faster.
Is factorising needed for quadratic equations and parabolas?
Yes. Factorising is very important for both quadratic equations and parabolas.
In quadratic equations, factorising helps students solve equations such as:
x² + 5x + 6 = 0
which becomes:
(x + 2)(x + 3) = 0
In parabolas, factorised form helps students find where the graph crosses the x-axis. This makes factorising a key skill for algebra and graphing in Stage 5 Mathematics.
