search
HomeComputer TutorialsComputer KnowledgeSolution formula of cubic equation of one variable!

Solution formula of cubic equation of one variable!

Jan 05, 2024 pm 07:50 PM
Solution formula of cubic equation of one variableThe root formula of a cubic equation of one variable? ? As long as it's publicRoot formula of cubic equation of one variable

What is the root formula of a cubic equation of one variable? ? Just the formula?

Solution to the root formula of a cubic equation of one variable

The root formula of a cubic equation of one variable cannot be obtained through ordinary deductive thinking, but the standard cubic equation can be simplified into a special form x^ by a method similar to the root formula of solving a quadratic equation. 3 px q=0. This method can help us solve the roots of a cubic equation of one variable more conveniently.

The solution to the solution formula of a cubic equation of one variable can only be obtained through inductive thinking. We can summarize based on the forms of root formulas of linear equations of one variable, quadratic equations of one variable and special higher-order equations, and thus obtain the form of the root formulas of cubic equations of one variable. The form obtained by induction is x = A^(1/3) B^(1/3), which is the sum of two open cubes. Then, we need to find the relationship between A and B and p and q. The specific method is as follows:

(1) Cube both sides of x=A^(1/3) B^(1/3) at the same time to get

(2)x^3=(A B) 3(AB)^(1/3)(A^(1/3) B^(1/3))

(3) Since x=A^(1/3) B^(1/3), (2) can be reduced to

x^3=(A B) 3(AB)^(1/3)x, you can get

by moving the terms

(4)x^3-3(AB)^(1/3)x-(A B)=0, compared with the cubic equation of one variable and the special type x^3 px q=0, it can be seen that

(5)-3(AB)^(1/3)=p,-(A B)=q, simplify to

(6)A B=-q,AB=-(p/3)^3

(7) In this way, the root formula of the cubic equation of one variable is actually transformed into the root formula of the quadratic equation, because A and B can be regarded as the two roots of the quadratic equation, and (6) is about The Vedic theorem of two roots of a quadratic equation of the form ay^2 by c=0, that is,

(8)y1 y2=-(b/a),y1*y2=c/a

(9) Comparing (6) and (8), we can set A=y1, B=y2, q=b/a,-(p/3)^3=c/a

(10) Since the root formula of a quadratic equation of type ay^2 by c=0 is

y1=-(b (b^2-4ac)^(1/2))/(2a)

y2=-(b-(b^2-4ac)^(1/2))/(2a)

can be changed to

(11)y1=-(b/2a)-((b/2a)^2-(c/a))^(1/2)

y2=-(b/2a) ((b/2a)^2-(c/a))^(1/2)

Substitute A=y1, B=y2, q=b/a,-(p/3)^3=c/a in (9) into (11) to get

(12)A=-(q/2)-((q/2)^2 (p/3)^3)^(1/2)

B=-(q/2) ((q/2)^2 (p/3)^3)^(1/2)

(13) Substitute A and B into x=A^(1/3) B^(1/3) to get

(14)x=(-(q/2)-((q/2)^2 (p/3)^3)^(1/2))^(1/3) (-(q/ 2) ((q/2)^2 (p/3)^3)^(1/2))^(1/3)

Equation (14) is only a real root solution of a three-dimensional equation of one variable. According to Vedic theorem, a cubic equation of one variable should have three roots. However, according to Vedic theorem, as long as one of the roots of a cubic equation of one variable is found, the other two roots will be easy. Out

Root formula of cubic equation of one variable

Solution to the root formula of a cubic equation of one variable

The root formula of a cubic equation of one variable cannot be obtained through ordinary deductive thinking, but the standard cubic equation can be simplified into a special form x^ by a method similar to the root formula of solving a quadratic equation. 3 px q=0. This method can help us solve the roots of a cubic equation of one variable more conveniently.

The solution to the solution formula of a cubic equation of one variable can only be obtained through inductive thinking. We can summarize based on the forms of root formulas of linear equations of one variable, quadratic equations of one variable and special higher-order equations, and thus obtain the form of the root formulas of cubic equations of one variable. The form obtained by induction is x = A^(1/3) B^(1/3), which is the sum of two open cubes. Then, we need to find the relationship between A and B and p and q. The specific method is as follows:

(1) Cube both sides of x=A^(1/3) B^(1/3) at the same time to get

(2)x^3=(A B) 3(AB)^(1/3)(A^(1/3) B^(1/3))

(3) Since x=A^(1/3) B^(1/3), (2) can be reduced to

x^3=(A B) 3(AB)^(1/3)x, you can get

by moving the terms

(4)x^3-3(AB)^(1/3)x-(A B)=0, compared with the cubic equation of one variable and the special type x^3 px q=0, it can be seen that

(5)-3(AB)^(1/3)=p,-(A B)=q, simplify to

(6)A B=-q,AB=-(p/3)^3

(7) In this way, the root formula of the cubic equation of one variable is actually transformed into the root formula of the quadratic equation, because A and B can be regarded as the two roots of the quadratic equation, and (6) is about The Vedic theorem of two roots of a quadratic equation of the form ay^2 by c=0, that is,

(8)y1 y2=-(b/a),y1*y2=c/a

(9) Comparing (6) and (8), we can set A=y1, B=y2, q=b/a,-(p/3)^3=c/a

(10) Since the root formula of a quadratic equation of type ay^2 by c=0 is

y1=-(b (b^2-4ac)^(1/2))/(2a)

y2=-(b-(b^2-4ac)^(1/2))/(2a)

can be changed to

(11)y1=-(b/2a)-((b/2a)^2-(c/a))^(1/2)

y2=-(b/2a) ((b/2a)^2-(c/a))^(1/2)

Substitute A=y1, B=y2, q=b/a,-(p/3)^3=c/a in (9) into (11) to get

(12)A=-(q/2)-((q/2)^2 (p/3)^3)^(1/2)

B=-(q/2) ((q/2)^2 (p/3)^3)^(1/2)

(13) Substitute A and B into x=A^(1/3) B^(1/3) to get

(14)x=(-(q/2)-((q/2)^2 (p/3)^3)^(1/2))^(1/3) (-(q/ 2) ((q/2)^2 (p/3)^3)^(1/2))^(1/3)

Equation (14) is only a real root solution of a three-dimensional equation of one variable. According to Vedic theorem, a cubic equation of one variable should have three roots. However, according to Vedic theorem, as long as one of the roots of a cubic equation of one variable is found, the other two roots will be easy. Out

The above is the detailed content of Solution formula of cubic equation of one variable!. For more information, please follow other related articles on the PHP Chinese website!

Statement
This article is reproduced at:Excel办公网. If there is any infringement, please contact admin@php.cn delete
Fixed: Windows Update May Have Automatically Replaced AMD - MiniToolFixed: Windows Update May Have Automatically Replaced AMD - MiniToolApr 18, 2025 am 12:51 AM

If your PC runs on an AMD chipset, you may receive the “Windows Update may have automatically replaced AMD” error message when using it. Don’t worry! This post from php.cn provides some solutions for you.

Microsoft Sway vs PowerPoint - Which One to Choose?Microsoft Sway vs PowerPoint - Which One to Choose?Apr 18, 2025 am 12:50 AM

What is Microsoft Sway and what is PowerPoint? These two are different presentation tools to facilitate people’s working and studying. So, what’s the difference between them and how choose between them. This post on php.cn Website will give you some

[Must-Know] Win 10 Offline Installer: Install Win 10 22H2 Offline[Must-Know] Win 10 Offline Installer: Install Win 10 22H2 OfflineApr 18, 2025 am 12:49 AM

Is there any Windows 10 offline installer for you to install the latest Windows 10 offline or without the internet? Of course, yes. php.cn Software shows you how to get a Windows 10 offline installer in this post.

Guide - How to Stop Expired Windows Server Auto-Shutdown?Guide - How to Stop Expired Windows Server Auto-Shutdown?Apr 18, 2025 am 12:48 AM

Some users report that they meet the Windows Server auto-shutdown issue after the license expires. This post from php.cn teaches you how to stop expired Windows Server auto-shutdown. Now, keep on your reading.

The File Can't Be Displayed in OneDrive - How to Resolve It?The File Can't Be Displayed in OneDrive - How to Resolve It?Apr 18, 2025 am 12:47 AM

Are you struggling with the “the file can’t be displayed” error when accessing the specific folder? Some users are complaining about this trouble and looking for useful measures. This article about the file can’t be displayed OneDrive from php.cn wil

Street Fighter 6 System Requirements – Are You Ready for It? - MiniToolStreet Fighter 6 System Requirements – Are You Ready for It? - MiniToolApr 18, 2025 am 12:46 AM

Street Fighter 6 is a fighting game published by Capcom and belongs to the Street Fighter franchise that has attracted a bunch of game fans. Some people hope to play this game on PCs. To do that, you need to meet some Street Fighter 6 system requirem

How to Show Drive Letter Before Drive Name Windows 10/11 - MiniToolHow to Show Drive Letter Before Drive Name Windows 10/11 - MiniToolApr 18, 2025 am 12:45 AM

Drive names (volume labels) are displayed before the drive letters by default in Windows. Do you know how to show drive letters before drive names? This post from php.cn tells you how to show drive letters first in File Explorer.

Exact Steps to Turn Off/On the Language Bar in Windows 10/11Exact Steps to Turn Off/On the Language Bar in Windows 10/11Apr 18, 2025 am 12:44 AM

The language bar is useful for a multilanguage Windows user. You can change the language by simply clicking. Some people want to turn off/on the language bar on the taskbar. Do you know how to do it? If you don’t know, read this php.cn post to find p

See all articles

Hot AI Tools

Undresser.AI Undress

Undresser.AI Undress

AI-powered app for creating realistic nude photos

AI Clothes Remover

AI Clothes Remover

Online AI tool for removing clothes from photos.

Undress AI Tool

Undress AI Tool

Undress images for free

Clothoff.io

Clothoff.io

AI clothes remover

AI Hentai Generator

AI Hentai Generator

Generate AI Hentai for free.

Hot Article

R.E.P.O. Energy Crystals Explained and What They Do (Yellow Crystal)
1 months agoBy尊渡假赌尊渡假赌尊渡假赌
R.E.P.O. Best Graphic Settings
1 months agoBy尊渡假赌尊渡假赌尊渡假赌
Will R.E.P.O. Have Crossplay?
1 months agoBy尊渡假赌尊渡假赌尊渡假赌

Hot Tools

Safe Exam Browser

Safe Exam Browser

Safe Exam Browser is a secure browser environment for taking online exams securely. This software turns any computer into a secure workstation. It controls access to any utility and prevents students from using unauthorized resources.

WebStorm Mac version

WebStorm Mac version

Useful JavaScript development tools

SAP NetWeaver Server Adapter for Eclipse

SAP NetWeaver Server Adapter for Eclipse

Integrate Eclipse with SAP NetWeaver application server.

MinGW - Minimalist GNU for Windows

MinGW - Minimalist GNU for Windows

This project is in the process of being migrated to osdn.net/projects/mingw, you can continue to follow us there. MinGW: A native Windows port of the GNU Compiler Collection (GCC), freely distributable import libraries and header files for building native Windows applications; includes extensions to the MSVC runtime to support C99 functionality. All MinGW software can run on 64-bit Windows platforms.

Atom editor mac version download

Atom editor mac version download

The most popular open source editor