Vijetove formulesu tvoj najbolji prijatelj kada radiš sa kvadratnim... Show more
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Updated Apr 17, 2026
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Vijetove formule: Veza između koeficijenata i rešenja
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Uvod u Vijetove formule
Zamisli da možeš da saznaš važne informacije o rešenjima kvadratne jednačine bez da je potpuno rešavaš - to je upravo ono što Vijetove formule čine! One povezuju koeficijente jednačine sa njenim rešenjima.
Za standardnu kvadratnu jednačinu ax² + bx + c = 0 (gde a ≠ 0), gde su x₁ i x₂ rešenja, imaš dve ključne formule:
• Zbir rešenja: x₁ + x₂ = -b/a
• Proizvod rešenja: x₁ · x₂ = c/a
💡 Pazi na minus! Najčešća greška je zaboravljanje minusa kod formule za zbir rešenja.
Zapamti: a, b i c su koeficijenti tvoje jednačine - a je uz x², b uz x, a c je slobodan član.
Detaljno objašnjenje i izvođenje
Da razumeš zašto Vijetove formule funkcionišu, pogledaj kako nastaju iz opšte formule za rešavanje kvadratnih jednačina. Ovo nije samo teorija - pomoći će ti da bolje zapamtiš!
Iz formule x₁,₂ = /2a, gde je D = b² - 4ac, možeš izvesti:
Za zbir rešenja: Kada sabereš x₁ i x₂, √D se pokrati i dobiješ x₁ + x₂ = -2b/2a = -b/a.
Za proizvod rešenja: Kada pomnojiš x₁ i x₂, koristiš razliku kvadrata i dobiješ x₁ · x₂ = c/a.
⚠️ Važna napomena: Jednačina mora biti u obliku ax² + bx + c = 0 pre primene formula!
Ove formule rade čak i kada rešenja nisu realna - zbir i proizvod će uvek biti realni brojevi.
Primena na složenije izraze
Ovde postaje stvarno zanimljivo! Možeš da transformišeš komplikovane izraze tako da zavise samo od zbira i proizvoda rešenja. Ovo je moćan alat za brže rešavanje.
Najčešće transformacije:
• Zbir kvadrata: x₁² + x₂² = ² - 2x₁x₂
• Zbir recipročnih vrednosti: 1/x₁ + 1/x₂ = /(x₁x₂)
• Zbir kubova: x₁³ + x₂³ = ³ - 3x₁x₂
Ključ je da prepoznaš obrasce i koristiš algebarske identitete poput kvadrata binoma i zbira kubova.
🎯 Pro tip: Uvek pokušaj da složeni izraz svedeš na kombinaciju x₁ + x₂ i x₁x₂!
Ovo ti štedi vreme jer ne moraš da rešavaš jednačinu - samo primeniš Vijetove formule i računaš dalje.
Rešeni primeri - deo 1
Primer 1: Za jednačinu 2x² - 7x + 5 = 0, pronađi zbir i proizvod rešenja.
Identifikuješ koeficijente: a = 2, b = -7, c = 5
Primenjuješ formule:
• Zbir: x₁ + x₂ = -(-7)/2 = 7/2
• Proizvod: x₁ · x₂ = 5/2 = 5/2
Primer 2: Za x² + 4x - 21 = 0, izračunaj x₁² + x₂².
Prvo nađeš: x₁ + x₂ = -4, x₁ · x₂ = -21
Zatim koristiš transformaciju: x₁² + x₂² = ² - 2x₁x₂ = (-4)² - 2(-21) = 16 + 42 = 58
💪 Zapamti: Nikad ne moraš da rešavaš kompletno jednačinu - formule rade sve umesto tebe!
Rešeni primeri - deo 2 i kreiranje jednačina
Primer 3: Sastavi kvadratnu jednačinu čija su rešenja za 3 veća od rešenja jednačine x² - 5x + 6 = 0.
Za početnu jednačinu: x₁ + x₂ = 5, x₁ · x₂ = 6
Nova rešenja su y₁ = x₁ + 3 i y₂ = x₂ + 3:
• Zbir novih: y₁ + y₂ = + 6 = 5 + 6 = 11
• Proizvod novih: y₁ · y₂ = = x₁x₂ + 3 + 9 = 6 + 15 + 9 = 30
Nova jednačina: y² - 11y + 30 = 0
🔥 Ključna veština: Kada znaš zbir i proizvod rešenja, možeš sastaviti jednačinu u obliku x² - (zbir)x + (proizvod) = 0!
Važne napomene: Pazi na znakove, uvek sređuj jednačinu u standardni oblik pre čitanja koeficijenata, i zapamti da formule važe čak i za kompleksna rešenja.
Kratak pregled za test
Evo svega što moraš da znaš napamet za uspešno rešavanje zadataka:
Osnovne formule:
• Standardni oblik: ax² + bx + c = 0
• Zbir rešenja: x₁ + x₂ = -b/a
• Proizvod rešenja: x₁ · x₂ = c/a
Kreiranje jednačine: Ako znaš rešenja x₁ i x₂, jednačina je x² - x + x₁x₂ = 0
Najvažnije transformacije:
• x₁² + x₂² = ² - 2x₁x₂
• 1/x₁ + 1/x₂ = /(x₁x₂)
✅ Test strategija: Uvek prvo identifikuj a, b, c, zatim pažljivo primeni formule - i ne zaboravi minus kod zbira!
Sa ovim znanjem možeš brzo i efikasno da rešavaš zadatke sa kvadratnim jednačinama bez dugotrajnih kalkulacija.
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 in Matematika
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Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
4.6/5
App Store
4.7/5
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 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 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 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 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
Vijetove formule: Veza između koeficijenata i rešenja
Vijetove formule su tvoj najbolji prijatelj kada radiš sa kvadratnim jednačinama! Ove formule ti omogućavaju da saznаš zbir i proizvod rešenja bez mučnog rešavanja cele jednačine - što je savršeno za brže rešavanje na testovima i kontrolnim.
Uvod u Vijetove formule
Zamisli da možeš da saznaš važne informacije o rešenjima kvadratne jednačine bez da je potpuno rešavaš - to je upravo ono što Vijetove formule čine! One povezuju koeficijente jednačine sa njenim rešenjima.
Za standardnu kvadratnu jednačinu ax² + bx + c = 0 (gde a ≠ 0), gde su x₁ i x₂ rešenja, imaš dve ključne formule:
• Zbir rešenja: x₁ + x₂ = -b/a
• Proizvod rešenja: x₁ · x₂ = c/a
💡 Pazi na minus! Najčešća greška je zaboravljanje minusa kod formule za zbir rešenja.
Zapamti: a, b i c su koeficijenti tvoje jednačine - a je uz x², b uz x, a c je slobodan član.
Sign up to see the contentIt's free!
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Improve your grades
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Detaljno objašnjenje i izvođenje
Da razumeš zašto Vijetove formule funkcionišu, pogledaj kako nastaju iz opšte formule za rešavanje kvadratnih jednačina. Ovo nije samo teorija - pomoći će ti da bolje zapamtiš!
Iz formule x₁,₂ = /2a, gde je D = b² - 4ac, možeš izvesti:
Za zbir rešenja: Kada sabereš x₁ i x₂, √D se pokrati i dobiješ x₁ + x₂ = -2b/2a = -b/a.
Za proizvod rešenja: Kada pomnojiš x₁ i x₂, koristiš razliku kvadrata i dobiješ x₁ · x₂ = c/a.
⚠️ Važna napomena: Jednačina mora biti u obliku ax² + bx + c = 0 pre primene formula!
Ove formule rade čak i kada rešenja nisu realna - zbir i proizvod će uvek biti realni brojevi.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
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Primena na složenije izraze
Ovde postaje stvarno zanimljivo! Možeš da transformišeš komplikovane izraze tako da zavise samo od zbira i proizvoda rešenja. Ovo je moćan alat za brže rešavanje.
Najčešće transformacije:
• Zbir kvadrata: x₁² + x₂² = ² - 2x₁x₂
• Zbir recipročnih vrednosti: 1/x₁ + 1/x₂ = /(x₁x₂)
• Zbir kubova: x₁³ + x₂³ = ³ - 3x₁x₂
Ključ je da prepoznaš obrasce i koristiš algebarske identitete poput kvadrata binoma i zbira kubova.
🎯 Pro tip: Uvek pokušaj da složeni izraz svedeš na kombinaciju x₁ + x₂ i x₁x₂!
Ovo ti štedi vreme jer ne moraš da rešavaš jednačinu - samo primeniš Vijetove formule i računaš dalje.
Sign up to see the contentIt's free!
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Improve your grades
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Rešeni primeri - deo 1
Primer 1: Za jednačinu 2x² - 7x + 5 = 0, pronađi zbir i proizvod rešenja.
Identifikuješ koeficijente: a = 2, b = -7, c = 5
Primenjuješ formule:
• Zbir: x₁ + x₂ = -(-7)/2 = 7/2
• Proizvod: x₁ · x₂ = 5/2 = 5/2
Primer 2: Za x² + 4x - 21 = 0, izračunaj x₁² + x₂².
Prvo nađeš: x₁ + x₂ = -4, x₁ · x₂ = -21
Zatim koristiš transformaciju: x₁² + x₂² = ² - 2x₁x₂ = (-4)² - 2(-21) = 16 + 42 = 58
💪 Zapamti: Nikad ne moraš da rešavaš kompletno jednačinu - formule rade sve umesto tebe!
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
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Rešeni primeri - deo 2 i kreiranje jednačina
Primer 3: Sastavi kvadratnu jednačinu čija su rešenja za 3 veća od rešenja jednačine x² - 5x + 6 = 0.
Za početnu jednačinu: x₁ + x₂ = 5, x₁ · x₂ = 6
Nova rešenja su y₁ = x₁ + 3 i y₂ = x₂ + 3:
• Zbir novih: y₁ + y₂ = + 6 = 5 + 6 = 11
• Proizvod novih: y₁ · y₂ = = x₁x₂ + 3 + 9 = 6 + 15 + 9 = 30
Nova jednačina: y² - 11y + 30 = 0
🔥 Ključna veština: Kada znaš zbir i proizvod rešenja, možeš sastaviti jednačinu u obliku x² - (zbir)x + (proizvod) = 0!
Važne napomene: Pazi na znakove, uvek sređuj jednačinu u standardni oblik pre čitanja koeficijenata, i zapamti da formule važe čak i za kompleksna rešenja.
Sign up to see the contentIt's free!
Access to all documents
Improve your grades
Join milions of students
Kratak pregled za test
Evo svega što moraš da znaš napamet za uspešno rešavanje zadataka:
Osnovne formule:
• Standardni oblik: ax² + bx + c = 0
• Zbir rešenja: x₁ + x₂ = -b/a
• Proizvod rešenja: x₁ · x₂ = c/a
Kreiranje jednačine: Ako znaš rešenja x₁ i x₂, jednačina je x² - x + x₁x₂ = 0
Najvažnije transformacije:
• x₁² + x₂² = ² - 2x₁x₂
• 1/x₁ + 1/x₂ = /(x₁x₂)
✅ Test strategija: Uvek prvo identifikuj a, b, c, zatim pažljivo primeni formule - i ne zaboravi minus kod zbira!
Sa ovim znanjem možeš brzo i efikasno da rešavaš zadatke sa kvadratnim jednačinama bez dugotrajnih kalkulacija.
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.
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Students love us — and so will you.
4.6/5
App Store
4.7/5
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 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 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 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 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