Algebraic expansion
Sections:
1) Revision notes
2) Practice questions
Revision notes
How to expand bracket(s)
How to expand a single bracket
$ 2(3x - 4) = \phantom{.} $ $ 6x - 8 $
How to expand brackets inside brackets
$ 18x - \frac{1}{2}[4x - 2(3 - x)] = \phantom{.} $ $ 15x + 3 $
![Expanding 18x − ½[4x − 2(3 − x)] step by step: inner bracket first to 4x − 6 + 2x, then 6x − 6, then 18x − 3x + 3, giving 15x + 3](/s/e-maths-expansion-nested.png)
How to expand the product of two brackets
How to expand by using algebraic identities
Square of a sum, (a + b)2
$ (a + b)^2 = \phantom{.} $ $ a^2 + 2ab + b^2 $
Example
$ (6x + 7y)^2 = \phantom{.} $ $ 36x^2 + 84xy + 49y^2 $
Square of a difference, (a - b)2
$ (a - b)^2 = \phantom{.} $ $ a^2 - 2ab + b^2 $
Example
$ (1 - 2xy)^2 = \phantom{.} $ $ 1 - 4xy + 4x^2 y^2$
Difference of two squares, (a + b)(a - b)
$ (a + b)(a - b) = \phantom{.} $ $ a^2 - b^2 $
Example
$ \left( \frac{1}{2}y + 3x\right)\left( \frac{1}{2}y - 3x\right) = \phantom{.} $ $ \frac{1}{4}y^2 - 9x^2 $
Practice questions
Expand and simplify, then compare coefficients
1. Given that $(x + 3)^2 + a(x - 2) = x^2 + bx + 5$ for all real values of $x$, find the value of $a$ and of $b$.
Answer: $ a = 2, b = 8 $
Expand and simplify, then use the result in a further problem
2(a) Expand and simplify $ (x - 5)^2 - (x - 2)(x + 2)$.
Answer: $ -10x + 29 $
2(b) Hence, evaluate $ 995^2 - 998 \times 1002 $. Show your workings clearly.
Answer: $ -9971 $
Expand and simplify, then show that the expression is even
3. Show that $3[n(2n - 1) + 4n] - (2n + 1)(3n - 2)$ is even for all positive integer values of $n$.
O Level past year questions on algebraic expansion
Fully worked, step-by-step solutions to these past-year questions (2016 to 2025) are in the O Level E Maths Solutions page. For 2015 and earlier, selected questions and their solutions are available to subscribers, linked individually in the table below.
| Year & paper | Comments |
|---|---|
| 2024 P1 Question 22 | |
| 2024 P1 Question 23 | (need to use Pythagoras' theorem) |
| 2023 P1 Question 12b | |
| 2023 P1 Question 15 | Show that the expression is a multiple of 3 |
| 2022 P1 Question 2b | |
| 2022 P1 Question 11b | |
| 2021 P1 Question 9b | |
| 2021 P1 Question 15 | Expand and simplify, then compare coefficients |
| 2020 P1 Question 19a | |
| 2018 P1 Question 5 | Show that the expression is a multiple of 3 |
| 2016 P1 Question 1a | |
| 2016 P1 Question 4 | Show that the expression is a multiple of 20 |
| 2013 P1 Question 1a | |
| 2012 P1 Question 11a | |
| 2009 P1 Question 1b | |
| 2008 P1 Question 15b | |
| 2007 P2 Question 2a | Expand and simplify, then use the result in a further problem (Link 🔒 Subscribers) |
| 2006 P1 Question 16a | Expand and simplify, then compare coefficients (Link 🔒 Subscribers) |
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