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Updated Mar 20, 2026
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Mastering Quadratic Equations: Factorization and Formula Techniques
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What Are Quadratic Equations?
Think of quadratic equations as algebra's next level challenge. Unlike simple linear equations that only have x, these always include an x² term, making them more interesting to solve. The highest power is always 2, which is what makes them "quadratic".
Every quadratic equation follows the same pattern: ax² + bx + c = 0. Getting your equation into this standard form is absolutely crucial before you start solving - it's like organising your desk before starting homework.
The letters a, b, and c are called coefficients - they're just the numbers in front of each term. Remember that 'a' can never be zero (otherwise it wouldn't be quadratic anymore!). Most quadratics have two solutions called roots, which are the x-values that make the equation true.
Quick tip: Roots and solutions mean exactly the same thing - don't let different terminology throw you off in exams!
Method 1: Solving by Factorising
This is often the fastest method, but only works when the quadratic can be factorised neatly. Think of it like breaking down a complex problem into smaller, manageable pieces.
Start by rearranging into standard form, then find the "guide number" by multiplying a and c together. You need two numbers that multiply to give this guide number AND add up to give b. Once you find them, rewrite the middle term using these numbers.
Now comes the clever bit: factorising by grouping. Group the first two terms and last two terms separately, take out common factors from each pair, and you should end up with matching brackets. Set each factor equal to zero and solve - that's your two solutions!
The key principle here is simple: if two things multiply to give zero, then one (or both) must be zero. So if = 0, then either x + 3 = 0 or x - 2 = 0.
Remember: This method is based on the zero product property - if the product equals zero, at least one factor must be zero.
Method 2: The Quadratic Formula
When factorising gets messy or impossible, the quadratic formula is your reliable backup. It works for every single quadratic equation, no exceptions. The best part? It's in your log tables, so you don't need to memorise it!
The formula is: x = / 2a. First, identify your a, b, and c values carefully - negative signs are especially tricky here. Substitute these into the formula using brackets to avoid sign errors.
Calculate the bit under the square root first, then split the calculation because of the ± symbol. You'll get two separate answers, which gives you both solutions. Watch out for questions asking for decimal places - that's usually a hint to use the formula!
The part under the square root is quite important. If it's negative, you can't find real solutions, so you'd write "no real roots" as your answer.
Exam tip: If a question asks for decimal places, it's almost always telling you to use the formula rather than factorising.
Worked Examples in Action
Let's see these methods in practice with real examples you might face in exams. For x² + 7x = -10, first rearrange to get x² + 7x + 10 = 0. The guide number is 1 × 10 = 10, and we need factors that add to 7.
Since 2 + 5 = 7 and 2 × 5 = 10, we rewrite as x² + 2x + 5x + 10 = 0. Grouping gives us x + 5 = 0, which factors to = 0. So x = -5 or x = -2.
For 2x² - 5x - 4 = 0, the decimal places hint tells us to use the formula. With a = 2, b = -5, c = -4, we substitute carefully: x = (5 ± √(25 + 32)) / 4 = (5 ± √57) / 4.
This gives us x = 3.14 and x = -0.64 (to two decimal places). Notice how the formula handles the messy numbers that would make factorising nearly impossible.
Pro tip: Always substitute your answers back into the original equation to check they work - it's a great way to catch mistakes!
Common Mistakes and Exam Strategy
The biggest mistake? Forgetting to rearrange to standard form first. If you see x² + 5x = 6, you MUST change it to x² + 5x - 6 = 0 before doing anything else. This trips up loads of students in exams.
Sign errors are another classic problem, especially with the formula. When b is negative, -b becomes positive. And remember (-5)² = 25, not -25! Take your time with substitution and use brackets to stay organised.
Don't forget that most quadratics have two solutions. The ± in the formula is there for a reason, and factorising should give you two brackets to solve. Missing a solution loses you marks.
Choose your method wisely: if the question asks for decimal places, use the formula. If the numbers look neat and simple, try factorising first. You can always switch methods if one isn't working out.
Final reminder: Check your answers by substituting back into the original equation - it only takes a minute and could save you valuable marks!
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the 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.
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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 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
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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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. 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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. 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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
Mastering Quadratic Equations: Factorization and Formula Techniques
Ever wondered why some algebra equations seem trickier than others? Quadratic equationsare the next step up from linear equations - they include an x² term and usually have two solutions instead of just one. Master these and you'll be... Show more
What Are Quadratic Equations?
Think of quadratic equations as algebra's next level challenge. Unlike simple linear equations that only have x, these always include an x² term, making them more interesting to solve. The highest power is always 2, which is what makes them "quadratic".
Every quadratic equation follows the same pattern: ax² + bx + c = 0. Getting your equation into this standard form is absolutely crucial before you start solving - it's like organising your desk before starting homework.
The letters a, b, and c are called coefficients - they're just the numbers in front of each term. Remember that 'a' can never be zero (otherwise it wouldn't be quadratic anymore!). Most quadratics have two solutions called roots, which are the x-values that make the equation true.
Quick tip: Roots and solutions mean exactly the same thing - don't let different terminology throw you off in exams!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Method 1: Solving by Factorising
This is often the fastest method, but only works when the quadratic can be factorised neatly. Think of it like breaking down a complex problem into smaller, manageable pieces.
Start by rearranging into standard form, then find the "guide number" by multiplying a and c together. You need two numbers that multiply to give this guide number AND add up to give b. Once you find them, rewrite the middle term using these numbers.
Now comes the clever bit: factorising by grouping. Group the first two terms and last two terms separately, take out common factors from each pair, and you should end up with matching brackets. Set each factor equal to zero and solve - that's your two solutions!
The key principle here is simple: if two things multiply to give zero, then one (or both) must be zero. So if = 0, then either x + 3 = 0 or x - 2 = 0.
Remember: This method is based on the zero product property - if the product equals zero, at least one factor must be zero.
Sign up to see the contentIt's free!
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Method 2: The Quadratic Formula
When factorising gets messy or impossible, the quadratic formula is your reliable backup. It works for every single quadratic equation, no exceptions. The best part? It's in your log tables, so you don't need to memorise it!
The formula is: x = / 2a. First, identify your a, b, and c values carefully - negative signs are especially tricky here. Substitute these into the formula using brackets to avoid sign errors.
Calculate the bit under the square root first, then split the calculation because of the ± symbol. You'll get two separate answers, which gives you both solutions. Watch out for questions asking for decimal places - that's usually a hint to use the formula!
The part under the square root is quite important. If it's negative, you can't find real solutions, so you'd write "no real roots" as your answer.
Exam tip: If a question asks for decimal places, it's almost always telling you to use the formula rather than factorising.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Worked Examples in Action
Let's see these methods in practice with real examples you might face in exams. For x² + 7x = -10, first rearrange to get x² + 7x + 10 = 0. The guide number is 1 × 10 = 10, and we need factors that add to 7.
Since 2 + 5 = 7 and 2 × 5 = 10, we rewrite as x² + 2x + 5x + 10 = 0. Grouping gives us x + 5 = 0, which factors to = 0. So x = -5 or x = -2.
For 2x² - 5x - 4 = 0, the decimal places hint tells us to use the formula. With a = 2, b = -5, c = -4, we substitute carefully: x = (5 ± √(25 + 32)) / 4 = (5 ± √57) / 4.
This gives us x = 3.14 and x = -0.64 (to two decimal places). Notice how the formula handles the messy numbers that would make factorising nearly impossible.
Pro tip: Always substitute your answers back into the original equation to check they work - it's a great way to catch mistakes!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Common Mistakes and Exam Strategy
The biggest mistake? Forgetting to rearrange to standard form first. If you see x² + 5x = 6, you MUST change it to x² + 5x - 6 = 0 before doing anything else. This trips up loads of students in exams.
Sign errors are another classic problem, especially with the formula. When b is negative, -b becomes positive. And remember (-5)² = 25, not -25! Take your time with substitution and use brackets to stay organised.
Don't forget that most quadratics have two solutions. The ± in the formula is there for a reason, and factorising should give you two brackets to solve. Missing a solution loses you marks.
Choose your method wisely: if the question asks for decimal places, use the formula. If the numbers look neat and simple, try factorising first. You can always switch methods if one isn't working out.
Final reminder: Check your answers by substituting back into the original equation - it only takes a minute and could save you valuable marks!
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
Where can I download the Knowunity app?
You can download the app in the Google Play Store and in the 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.
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4.7/5
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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 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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. 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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
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
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
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
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE Knowunity AI. 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 app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user