Factorising is about breaking an expression down into simpler terms, making it easier to work with. Factorising an expression is the opposite of the process of expanding brackets. It is important when simplifying algebraic expressions, solving equations and understanding graphs.
This article will explain the methods for factorising expressions. This includes single brackets, quadratics and the difference of two squares. Examples and quiz questions are included to test your understanding. It is aimed at students undertaking GCSE Maths revision, and the exam boards it is suitable for include AQA and OCR.
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Factorising quadratics (a = 1)
A quadratic expression contains an x2 term, an x term and a constant. For example, x2 + 5x + 6x. To factorise a quadratic equation when the coefficient in front of x2 is 1 (which means a = 1), you need to find two numbers. These numbers multiply together to equal the constant term and also add up to the middle coefficient.
For instance, x2 + 7x + 12:
- We need to find two numbers that multiply together to give 12 and add together to give 7.
- The possible pairs are (1, 12), (2, 6) and (3, 4).
- 3 and 4 fit because 3 x 4 = 12 and 3 + 4 = 7.
- The expression factorises into (x+3)(x+4).
The signs in the quadratic expression are important. If the expression is x2 - 9x + 20, we need numbers that multiply to 20 and also add together to -9. The pair is -4 and -5 (-4 x -5 = 20 | -4 + -5 = -9). The expression factorises to (x - 4)(x - 5).
It's a good idea to double-check your factorisation by expanding the brackets again. The answer is correct if you end up back at the original quadratic.
1
How do you factorise x2 + 8x + 15?
Factorising quadratics (a isn't equal to 1)
The process of factorising quadratic equations differs when the number in front of x2 is greater than 1. This method requires an extra step, but remains straightforward.
For example, 2x2 + 7x + 3:
- Multiply the coefficient of x2 (a) by the constant term (c) - We need to multiply 2 × 3 = 6.
- Find two numbers that multiply to equal the constant term (6) and add together to give the middle coefficient (7). These numbers are 6 and 1.
- Split the middle term (7x) into two parts - We use the numbers 2x2 + 6x + x + 3.
- Group the terms in pairs - if we factor out 2x from 2x2 + 6x, we get 2x(x + 3). If we factor out 1 from x + 3, we are left with 1(x+3).
- Factor out the common bracket - Both groups now share (x+3). If we factor this out, we are left with (x+3)(2x+1).
- Expand the brackets to confirm your answer is correct.
This process becomes intuitive with practice. It's arguably the most effective method for breaking down complex quadratics.
2
How do you factorise 3x2 + 11x + 6?
Difference of two squares
You are expected to find the difference between two squares in your GCSE exams. The method is very direct and requires few steps. Bear in mind that we only use it in specific circumstances.
As a quick reminder, a square number is the product of a number that is multiplied by itself. For example, 4 is a square (2 x 2) and is written as 22. 9 is another square (3 x 3) and is written as 32.
You can see a difference of squares when there are two perfect squares separated by a minus sign, such as a² - b². For instance, x² - 9 is a difference of squares because x2 is a square and so is 9, as we discussed above.
This is the factorising pattern:
a2 - b2 = (a + b)(a - b)
In the previous example, x² – 9 becomes (x + 3)(x - 3). This is the answer - there are no further steps required.
Another example is 4x² - 25. Both terms are perfect squares: 4x² is (2x)² and 25 is 5² (5 x 5 = 25). If we use the formula above, we factorise it as (2x + 5)(2x - 5).
This method only works where there is a subtraction between the squares. We can't factorise x² + 9 because it’s a sum of squares. You aren't expected to factorise these numbers for GCSE.
If you see two square terms and a minus sign between them, the question is pointing you towards applying the "difference of squares" method.
3
What is the correct factorisation of x² - 16?
Final thoughts on factorising expressions - GCSE Maths
We have explored the logical steps to take when factorising expressions. A daunting question is much more approachable when you break it down into stages, much like the process of factorising itself. The key is to practice, including how to find a common factor, split up a quadratic, and find a difference of squares.
For further reading, you can test your understanding with past paper questions on expanding and factorising by MathsGenie. You can also learn about the related topic of expanding brackets by ThirdSpaceLearning.
If you need help with this topic, TeachTutti has verified GCSE Maths tutors to help you revise how to simplify algebra expressions and factorising in general. Tutors will tailor lessons to your specific needs, such as working through past paper factorising questions.
Glossary
- Coefficient - A number placed before a variable, e.g. "3" in 3x.
- Constant - A fixed number on its own, without a variable, such as 4 or 9.
- Expression - A phrase that contains numbers, variables and operation symbols, e.g. 5x + 3).
- Factorise - When you rewrite an expression as a product of simpler expressions. Normally, this means using brackets.
- Quadratic expression - An algebraic expression with x2 as the highest power. For instance, x2 + 3x + 2).
- Term - A single component of an expression. It is separated by plus or minus signs, e.g. 5x, 3 and 2y2 are all terms.
- Variable - A letter that represents an unknown/changing quantity, such as x or y.
