Quadratic expressions might look intimidating at first, but they're actually...
Algebra 187 views·Updated 24 Aug 2026·4 pagesMaster Quadratics: Factorisation and Solutions - GCSE/IGCSE Guide
Neil Trivedi - MyEdSpace@neildoesmaths
1 / 4
1
of 4
Getting Started with Factorising
Factorising quadratics means rewriting expressions like x² - 3x - 28 as two brackets multiplied together. It's like finding the "ingredients" that make up the original expression.
The key is finding two numbers that multiply to give you the constant term (the number without x). For x² - 3x - 28, you need pairs that multiply to make 28: (1,28), (2,14), or (4,7).
Next, check which pair adds or subtracts to give you the middle coefficient. Since 4 - 7 = -3, you get x + 4$$x - 7. Always double-check by expanding your answer!
Quick Tip: Start by listing all factor pairs of the constant term - this saves time and prevents mistakes.
2
of 4
Difference of Two Squares (DOTS)
When you spot something like x² - 64, you're looking at a difference of two squares - a special pattern that's much easier to factorise than regular quadratics.
Both terms must be perfect squares with a minus sign between them. The answer always follows the pattern: x + 8$$x - 8.
This works because when you expand these brackets, the middle terms cancel each other out perfectly. No need for trial and error - just take the square root of each term and use opposite signs.
Remember: DOTS always gives you a + b$$a - b where a² and b² are your original terms.
3
of 4
Harder Quadratics and Simplifying Fractions
When the coefficient of x² isn't 1 (like in 2x² + 3x + 1), factorising becomes trickier but follows the same logic. Start with factors of the x² term: here it's 2x and x.
For the constant term, find factor pairs (1 and 1 in this case), then test different arrangements in brackets. Expand the middle terms to check: 2x + 1$$x + 1 gives 2x + x = 3x ✓
Factorising helps simplify algebraic fractions too. When you have quadratics in both numerator and denominator, factorise both and cancel common factors. It's like cancelling fractions but with algebra!
Pro Strategy: When arranging factors, try the combination where numbers are closest together first - it often works faster.
4
of 4
Solving Quadratic Equations
Once you've factorised a quadratic equation like 15x² + x - 2 = 0, solving becomes straightforward. If two brackets multiply to give zero, one of them must equal zero.
For 15x², try factor pairs like (5x, 3x) rather than (15x, x) - closer numbers usually work better. After some trial with arranging factors, you get 5x + 2$$3x - 1 = 0.
This means either 5x + 2 = 0 or 3x - 1 = 0, giving you x = -2/5 or x = 1/3. Always give both solutions unless the question asks for just one.
Check Yourself: Substitute your answers back into the original equation - both should make it equal zero.
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Algebra 187 views·Updated 24 Aug 2026·4 pagesMaster Quadratics: Factorisation and Solutions - GCSE/IGCSE Guide
Neil Trivedi - MyEdSpace@neildoesmaths
Quadratic expressions might look intimidating at first, but they're actually quite manageable once you know the tricks. Think of factorising as reverse engineering - you're breaking down a complex expression into simpler building blocks that multiply together.
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Getting Started with Factorising
Factorising quadratics means rewriting expressions like x² - 3x - 28 as two brackets multiplied together. It's like finding the "ingredients" that make up the original expression.
The key is finding two numbers that multiply to give you the constant term (the number without x). For x² - 3x - 28, you need pairs that multiply to make 28: (1,28), (2,14), or (4,7).
Next, check which pair adds or subtracts to give you the middle coefficient. Since 4 - 7 = -3, you get x + 4$$x - 7. Always double-check by expanding your answer!
Quick Tip: Start by listing all factor pairs of the constant term - this saves time and prevents mistakes.
2
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Difference of Two Squares (DOTS)
When you spot something like x² - 64, you're looking at a difference of two squares - a special pattern that's much easier to factorise than regular quadratics.
Both terms must be perfect squares with a minus sign between them. The answer always follows the pattern: x + 8$$x - 8.
This works because when you expand these brackets, the middle terms cancel each other out perfectly. No need for trial and error - just take the square root of each term and use opposite signs.
Remember: DOTS always gives you a + b$$a - b where a² and b² are your original terms.
3
of 4Sign up to see the content. It's free!
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Harder Quadratics and Simplifying Fractions
When the coefficient of x² isn't 1 (like in 2x² + 3x + 1), factorising becomes trickier but follows the same logic. Start with factors of the x² term: here it's 2x and x.
For the constant term, find factor pairs (1 and 1 in this case), then test different arrangements in brackets. Expand the middle terms to check: 2x + 1$$x + 1 gives 2x + x = 3x ✓
Factorising helps simplify algebraic fractions too. When you have quadratics in both numerator and denominator, factorise both and cancel common factors. It's like cancelling fractions but with algebra!
Pro Strategy: When arranging factors, try the combination where numbers are closest together first - it often works faster.
4
of 4Sign up to see the content. It's free!
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- Improve your grades
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By signing up you accept Terms of Service and Privacy Policy
Solving Quadratic Equations
Once you've factorised a quadratic equation like 15x² + x - 2 = 0, solving becomes straightforward. If two brackets multiply to give zero, one of them must equal zero.
For 15x², try factor pairs like (5x, 3x) rather than (15x, x) - closer numbers usually work better. After some trial with arranging factors, you get 5x + 2$$3x - 1 = 0.
This means either 5x + 2 = 0 or 3x - 1 = 0, giving you x = -2/5 or x = 1/3. Always give both solutions unless the question asks for just one.
Check Yourself: Substitute your answers back into the original equation - both should make it equal zero.
We thought you’d never ask...
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
You can download the app from Google Play Store and Apple App Store.
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Students love us — and so will you.
4.6/5App Store
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The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan SiOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha KlichAndroid user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
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