Quadratic equations might seem scary at first, but they're actually... Show more
English Literature
Biology
SociologySign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
924
•
7 Jan 2026
•
Mastering Quadratic Equations
issy
@issy.studies
1 / 6
Factorising Simple Quadratic Equations
The easiest way to solve quadratic equations is by factorising when the coefficient of x² is 1. For an equation like x² + 5x + 6 = 0, you need to find two numbers that multiply to give c (the constant term) and add to give b (the coefficient of x).
Take x² + 5x + 6 = 0 as an example. List all the factor pairs of 6: (1,6), (2,3), (-1,-6), (-2,-3). The pair that adds up to 5 is 2 and 3, so you get = 0.
Once you've factorised, set each bracket equal to zero: = 0 gives x = -2, and = 0 gives x = -3. Always check your answers by substituting back into the original equation!
Top Tip: When factorising, remember that if c is positive and b is positive, both numbers in your factor pair will be positive. If c is positive and b is negative, both numbers will be negative.
Factorising When the Coefficient of x² Isn't 1
When you have something like 2x² + 7x + 6 = 0, factorising gets a bit trickier but follows the same principle. You need to find two numbers that multiply to give ac (a times c) and add to give b.
For 2x² + 7x + 6 = 0, multiply a and c: 2 × 6 = 12. Find factor pairs of 12 that add to 7. The pair 3 and 4 works perfectly.
Split the middle term using these numbers, then factorise by grouping. You'll end up with = 0, giving you x = -2 or x = -3/2. This method takes practice, but once you get the hang of it, it becomes second nature.
Quick Check: Always multiply out your brackets to verify you get back to the original equation - it's a great way to catch silly mistakes!
More Complex Factorising Examples
These trickier quadratic equations follow the same pattern but with bigger numbers. For 6x² + 11x + 3 = 0, multiply a and c: 6 × 3 = 18. You need factor pairs of 18 that add to 11.
The factors of 18 include (2,9) and (3,6). Since 2 + 9 = 11, you split 11x into 2x + 9x. This gives you 6x² + 2x + 9x + 3, which factors as = 0.
Solving gives x = -3/2 or x = -1/3. The same method works for any quadratic - find the right factor pair, split the middle term, then factorise by grouping. Practice with different examples helps you spot the patterns more quickly.
Memory Aid: Write down all factor pairs systematically - don't try to work them out in your head, as you might miss the right combination!
Building Confidence with Factorising
Let's tackle 4x² + 19x + 12 = 0 using the same reliable method. First, calculate ac: 4 × 12 = 48. Now list the factor pairs of 48 systematically.
The factors include (1,48), (2,24), (3,16), (4,12), and (6,8). Since we need them to add to 19, the pair (3,16) works perfectly because 3 + 16 = 19.
Split 19x into 3x + 16x, giving 4x² + 3x + 16x + 12. Factor by grouping to get = 0. This gives solutions x = -4 or x = -3/4. See how the same method works every time?
Confidence Booster: Once you've mastered this systematic approach, you can solve any factorable quadratic equation - no matter how complex it looks!
Completing the Square Method
Completing the square is brilliant when factorising doesn't work easily. For x² + 10x + 10, take half the coefficient of x (which gives 5), then write ² - 25 + 10 = ² - 15.
To solve ² - 15 = 0, rearrange to get ² = 15, then take the square root: x + 5 = ±√15. Therefore x = -5 ± √15.
When the coefficient of x² isn't 1, like in 2x² - 6x - 3, factor out the 2 first: 2 - 3. Complete the square inside the brackets: 2² - 2(9/4) - 3 = 2² - 15/2.
Why It's Useful: Completing the square also tells you the minimum or maximum point of the quadratic graph - super handy for coordinate geometry questions!
The Quadratic Formula - Your Ultimate Backup
The quadratic formula x = /2a works for absolutely any quadratic equation. When factorising gets messy or completing the square is awkward, this formula saves the day.
For any equation in the form ax² + bx + c = 0, just substitute your values. Take 2x² + 6x + 3 = 0: a = 2, b = 6, c = 3. Plugging into the formula gives x = (-6 ± √(36 - 24))/4 = (-6 ± √12)/4.
This simplifies to x = (-6 ± 2√3)/4 = (-3 ± √3)/2. You can use your calculator to get decimal answers: approximately -0.63 and -2.37. The quadratic formula never lets you down!
Pro Tip: Learn to recognise when b² - 4ac (the discriminant) is negative - this means there are no real solutions, which is important information for your answer!
We thought you’d never ask...
What is the Knowunity AI companion?
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.
Where can I download the Knowunity app?
You can download the app from Google Play Store and Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
Most popular content: Quadratic Equation
Most popular content in Maths
Most popular content
Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
4.9/5
App Store
4.8/5
Google Play
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 S
iOS 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 Klich
Android 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.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
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 S
iOS 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 Klich
Android 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.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Mastering Quadratic Equations
issy
@issy.studies
Quadratic equations might seem scary at first, but they're actually quite straightforward once you know the tricks. There are several brilliant methods to solve them, from simple factorising to completing the square and using the quadratic formula.
Factorising Simple Quadratic Equations
The easiest way to solve quadratic equations is by factorising when the coefficient of x² is 1. For an equation like x² + 5x + 6 = 0, you need to find two numbers that multiply to give c (the constant term) and add to give b (the coefficient of x).
Take x² + 5x + 6 = 0 as an example. List all the factor pairs of 6: (1,6), (2,3), (-1,-6), (-2,-3). The pair that adds up to 5 is 2 and 3, so you get = 0.
Once you've factorised, set each bracket equal to zero: = 0 gives x = -2, and = 0 gives x = -3. Always check your answers by substituting back into the original equation!
Top Tip: When factorising, remember that if c is positive and b is positive, both numbers in your factor pair will be positive. If c is positive and b is negative, both numbers will be negative.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Factorising When the Coefficient of x² Isn't 1
When you have something like 2x² + 7x + 6 = 0, factorising gets a bit trickier but follows the same principle. You need to find two numbers that multiply to give ac (a times c) and add to give b.
For 2x² + 7x + 6 = 0, multiply a and c: 2 × 6 = 12. Find factor pairs of 12 that add to 7. The pair 3 and 4 works perfectly.
Split the middle term using these numbers, then factorise by grouping. You'll end up with = 0, giving you x = -2 or x = -3/2. This method takes practice, but once you get the hang of it, it becomes second nature.
Quick Check: Always multiply out your brackets to verify you get back to the original equation - it's a great way to catch silly mistakes!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
More Complex Factorising Examples
These trickier quadratic equations follow the same pattern but with bigger numbers. For 6x² + 11x + 3 = 0, multiply a and c: 6 × 3 = 18. You need factor pairs of 18 that add to 11.
The factors of 18 include (2,9) and (3,6). Since 2 + 9 = 11, you split 11x into 2x + 9x. This gives you 6x² + 2x + 9x + 3, which factors as = 0.
Solving gives x = -3/2 or x = -1/3. The same method works for any quadratic - find the right factor pair, split the middle term, then factorise by grouping. Practice with different examples helps you spot the patterns more quickly.
Memory Aid: Write down all factor pairs systematically - don't try to work them out in your head, as you might miss the right combination!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Building Confidence with Factorising
Let's tackle 4x² + 19x + 12 = 0 using the same reliable method. First, calculate ac: 4 × 12 = 48. Now list the factor pairs of 48 systematically.
The factors include (1,48), (2,24), (3,16), (4,12), and (6,8). Since we need them to add to 19, the pair (3,16) works perfectly because 3 + 16 = 19.
Split 19x into 3x + 16x, giving 4x² + 3x + 16x + 12. Factor by grouping to get = 0. This gives solutions x = -4 or x = -3/4. See how the same method works every time?
Confidence Booster: Once you've mastered this systematic approach, you can solve any factorable quadratic equation - no matter how complex it looks!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Completing the Square Method
Completing the square is brilliant when factorising doesn't work easily. For x² + 10x + 10, take half the coefficient of x (which gives 5), then write ² - 25 + 10 = ² - 15.
To solve ² - 15 = 0, rearrange to get ² = 15, then take the square root: x + 5 = ±√15. Therefore x = -5 ± √15.
When the coefficient of x² isn't 1, like in 2x² - 6x - 3, factor out the 2 first: 2 - 3. Complete the square inside the brackets: 2² - 2(9/4) - 3 = 2² - 15/2.
Why It's Useful: Completing the square also tells you the minimum or maximum point of the quadratic graph - super handy for coordinate geometry questions!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
The Quadratic Formula - Your Ultimate Backup
The quadratic formula x = /2a works for absolutely any quadratic equation. When factorising gets messy or completing the square is awkward, this formula saves the day.
For any equation in the form ax² + bx + c = 0, just substitute your values. Take 2x² + 6x + 3 = 0: a = 2, b = 6, c = 3. Plugging into the formula gives x = (-6 ± √(36 - 24))/4 = (-6 ± √12)/4.
This simplifies to x = (-6 ± 2√3)/4 = (-3 ± √3)/2. You can use your calculator to get decimal answers: approximately -0.63 and -2.37. The quadratic formula never lets you down!
Pro Tip: Learn to recognise when b² - 4ac (the discriminant) is negative - this means there are no real solutions, which is important information for your answer!
We thought you’d never ask...
What is the Knowunity AI companion?
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.
Where can I download the Knowunity app?
You can download the app from Google Play Store and Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
7
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
Mock Exam
Quiz
Flashcards
Essay
Similar content
Quadratic Formula Insights
Explore essential GCSE and A Level mathematics concepts on completing the square. This resource includes the key formula, a proof of the quadratic formula, and practice questions to enhance your understanding of quadratic equations.
Maths12
GCSE Higher Maths Practice
Enhance your skills with this GCSE Higher Maths practice worksheet, featuring a variety of problems including geometry, algebra, and calculator techniques. Perfect for exam preparation, this resource covers key concepts such as triangular numbers, prime numbers, LCM, and more. Ideal for students aiming for top grades in their GCSE Maths exam.
Maths9
Quadratic Equation Techniques
Master the methods for solving quadratic equations, including the quadratic formula, completing the square, and factorization. This summary provides essential tips and identities to help you tackle various quadratic problems effectively. Ideal for students preparing for exams or needing a quick reference.
Maths10
National 5 Maths Skills Checklist
Comprehensive skills checklist for National 5 Maths covering key topics such as scientific notation, fractions, percentages, algebraic manipulation, trigonometry, quadratic equations, and more. Ideal for exam preparation and revision, this resource helps students identify essential mathematical concepts and practice their skills effectively.
MathsS4
Mastering Quadratic Functions
Explore key concepts of quadratic functions, including the quadratic formula, vertex form, and axis of symmetry. This comprehensive guide features example calculations and problem-solving strategies to enhance your understanding and revision for Nat 5 Maths.
MathsS3
Stationary & Inflection Points
Explore the concepts of stationary points and points of inflection in calculus. This resource covers the first and second derivative tests, critical points, and concavity analysis, providing clear examples and explanations for A-level Maths students.
Maths12
Most popular content: Quadratic Equation
Most popular content in Maths
Most popular content
Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
4.9/5
App Store
4.8/5
Google Play
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 S
iOS 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 Klich
Android 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.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
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 S
iOS 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 Klich
Android 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.
Anna
iOS user
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user