3 Feb 2025

How to Factorise Quadratics: A Simple Guide That GCSE Students Actually Understand

Robin Gibbons

Students who receive proper guidance in mathematics improve their grades by 1.19 on average. This improvement represents nearly half a grade better than those without support!

Many students find mathematical concepts challenging, especially when they learn how to factorise quadratics. The formula ax² + bx + c might look simple at first glance, but converting it to the form (x + a)(x + b) feels like solving a puzzle without seeing the picture.

We know this can be difficult. Our simple guide to factorising quadratic expressions will help you succeed. Each step breaks down into clear, easy-to-follow instructions that work for both simple quadratic equations and complex problems with coefficients.

Are you ready to become skilled at factorising quadratics? Let's start with the fundamentals!

Understanding Quadratic Expressions

A quadratic expression takes the form ax² + bx + c, where 'a' is non-zero, and a, b, and c are real numbers [1]. The word 'quadratic' has its roots in the Latin word 'quadratus', which means square and refers to the x² term [2].

Each part of a quadratic expression plays a unique role. The term ax² is the quadratic term, bx is the linear term, and c is the constant term [3]. To cite an instance, the expression 8x² + 7x - 1 has values a = 8, b = 7, and c = -1 [4].

Quadratic expressions matter because they show the path of a parabola - a U-shaped curve that opens upward or downward [1]. These expressions help mathematicians and scientists model ground scenarios, like objects moving under constant acceleration [3].

The main difference between quadratic expressions and equations stands out clearly. An expression like 5x² - 3x + 7 exists by itself, while an equation has an equals sign, such as 3x² + 25x - 18 = 0 [5]. This becomes vital as you learn to factorise quadratics.

Knowing how to spot and handle quadratic expressions are the foundations for advanced math concepts, including solving quadratic equations and graphing parabolas [1].

Master the Basic Factorising Steps

Let's become skilled at the basic steps of factorising quadratics with a clear method. Success in factorisation depends on finding two numbers that work together in a specific way.

The first step in factorising a quadratic expression needs the standard form ax² + bx + c. Next, find two numbers that multiply to give ac (the product of the coefficient of x² and the constant term) and add to give b (the coefficient of x) [6].

Here's the systematic process to factorise quadratics:

Find the Product: Multiply a and c to get ac
Identify Factors: List the factor pairs of ac
Check Addition: Find the pair that adds to give b
Write Brackets: Place x at the start of each bracket
Add Numbers: Insert the chosen factors with appropriate signs

You should always verify your answer by expanding the brackets. To cite an instance, see when (x + 4)(x - 1) expands to x² + 3x - 4, you know your factorisation is correct [6].

On top of that, it helps to recognise certain patterns that make factorising easier. Watch for special cases like the difference of two squares (x² - a²), which always factorises to (x + a)(x - a) [7]. So, x² - 16 factorises to (x + 4)(x - 4).

Note that factorising reverses the process of expanding brackets. If expanding (x + a)(x + b) gives x² + (a + b)x + ab, factorising works backwards from the expanded form to find the original brackets [6].

Common Factorising Patterns

Students find factorising quadratics easier when they spot special patterns. Two patterns form the foundations of what every GCSE student should become skilled at.

The first pattern is the difference of squares, which shows up in quadratic expressions with the form x² - a². Both terms need to be perfect squares, and subtraction must connect them [7]. To cite an instance, x² - 16 breaks down to (x + 4)(x - 4), because 16 equals 4² [6].

The perfect square trinomial stands as the second pattern. This pattern follows a simple structure: x² ± 2xy + y², which breaks down to (x ± y)². To name just one example, see x² + 10x + 25 - it factors to (x + 5)², as 25 equals 5² [8].

Common factors can sometimes hide these patterns. Take 98x³ - 2x. The first step pulls out 2x to get 2x(49x² - 1). Now you can see the difference of squares clearly in the brackets [9].

These key patterns will help you:

Difference of squares: a² - b² = (a + b)(a - b)
Perfect square trinomial: a² ± 2ab + b² = (a ± b)²

Students who spot these patterns make fewer mistakes and save time. Practise will help you recognise these patterns quickly, and factorising quadratics becomes easier [7].

Conclusion

Factorising quadratics might seem scary at first. The good news is that anyone can become skilled at this essential mathematical skill with proper understanding and practise. Students need to recognise the components of quadratic expressions and learn systematic steps for factorisation.

Students who learn these fundamental concepts can handle more complex mathematical problems easily. Spotting patterns like the difference of squares and perfect square trinomials makes solving quadratic expressions a lot easier.

Success comes from consistent practise and using these methods regularly. Each quadratic expression gives you a chance to build your mathematical skills. Your confidence will grow as you work with these expressions, and factorising will become second nature.

Regular practise with these techniques will improve your mathematical abilities. Mathematics works like building blocks - each concept you master creates a stronger foundation to learn more advanced topics.

FAQs

Q1. How do I factorise quadratic expressions for GCSE maths? To factorise quadratics, write the expression in the form ax² + bx + c. Find two numbers that multiply to give ac and add to give b. Use these numbers to split the middle term, then factor by grouping. Always check your answer by expanding the brackets.

Q2. What are the key steps in factorising quadratic equations? The key steps are: 1) Identify the values of a, b, and c in the quadratic expression. 2) Find two numbers that multiply to ac and add to b. 3) Rewrite the middle term using these numbers. 4) Factor by grouping. 5) Write the final factorised form (x + p)(x + q).

Q3. Are there any special patterns that make factorising quadratics easier? Yes, two common patterns are the difference of squares (a² - b² = (a + b)(a - b)) and perfect square trinomials (a² ± 2ab + b² = (a ± b)²). Recognising these patterns can significantly simplify the factorisation process.

Q4. How can I improve my quadratic factorisation skills? Practise regularly with a variety of quadratic expressions. Start with simpler examples and gradually move to more complex ones. Pay attention to special patterns and always verify your answers by expanding the factorised form.

Q5. Why is factorising quadratics important in mathematics? Factorising quadratics is crucial for solving quadratic equations, graphing parabolas, and understanding more advanced mathematical concepts. It's a fundamental skill that helps in modelling real-world scenarios and forms the basis for higher-level mathematics.