Get 20% Off + $20 Signup Bonus ! Limited Time, Hurry !
Get 20% Off + $20 Signup Bonus ! Limited Time, Hurry !
Securing Higher Grades Costing Your Pocket? Book Your Assignment at The Lowest Price Now!
loader
Add File

Error goes here

Files Missing!

Please upload all relevant files for quick & complete assistance.

Save Time & Excel Academics

Share your requirements & connect with top tutor !

watch
question
We will use e-mail only for:

arrow Communication regarding your orders

arrow To send you invoices, and other billing info

arrow To provide you with information of offers and other benefits

Add File

Files Missing!

Please upload all relevant files for quick & complete assistance.

2,019,747

Orders

4.9/5

Overall Rating

5,178

Experts

How to Solve a Cubic Equation : ax3+ bx2+cx+d= 0

UserMark time23 December,2014

Learn To Solve Cubic Equations

In mathematical terms, all cubic equations have either one root or three real roots. The general cubic equation is,

ax3+ bx2+ cx+d= 0

The coefficients of a, b, c, and d are real or complex numbers with a not equals to zero (a ≠ 0). It must have the term x3   in it, or else it will not be a cubic equation. But any or all of b, c and d can be zero.

The examples of cubic equations are,

  1. 3x3+ 3x2 + x– b=0
  2. 4x3+ 57=0
  3. 1.x3+ 9x=0   or x3+ 9x=0

Note:  a or the coefficient before x3 (that is highlighted) is not equal to 0.The highest power of variable x in the equation is 3.

How to Solve Cubic Equations

Ways of solving cubic equation

Unlike quadratic equation which may have no real solution; a cubic equation always has at least one real root. The prior strategy of solving a cubic equation is to reduce it to a quadratic equation, and then solve the quadratic by usual means, either by factorizing or using a formula.

Always try to find the solution of cubic equations with the help of the general equation,

ax3+ bx2+ cx+d= 0

A cubic equation should, therefore, must be re-arranged into its standard form,

For example,

x2+ 4x-1 =  6/x

Step 1

You can see the equation is not written in standard form, you need to multiply the ‘x’ to eliminate the fraction and get cubic equation, after doing so, you will end up with

x (x2+ 4x-1) = 6

x3+ 4x2– x = 6

Step 2

Then you subtract 6 from both sides in order to get ‘0’ on the right side, so you will come up,

x3+ 4x2– x- 6 = 0

Solving cubic equations with the help of factor theorem

What is factor theorem? If you divide a polynomial p(x) by a factor x – a of that polynomial, then, you will end up with zero as the remainder,

p(x) = (x – a) q(x) + r(x)

If x – a is indeed a factor of p(x), then the remainder after division by x – a will be zero.

p(x) = (x – a)q (x)

Here is a problem,

x3– 5x2– 2x+24 = 0

With x= – 2 a solution.

Factor theorem says that if  x = – 2 is a solution of this equation, then x+2 is a factor of this whole expression. So,

x3– 5x2– 2x+24 = 0

can be written in the form,

(x+2)( x2+ ax+b)=0  

Where a and b are numbers,

You need to find out what are the values of a and b here by using synthetic division. 

Step 1

First, you need to look at the coefficients of the original cubic equation, which are 1, -5, -2 and 24.

To the right of the vertical you need to write down the known root, x = -2

Step 2

Now multiply number (1) that just brought down, by the known root -2, as a result is -2, you mention the result in the other line, like

Step 3

The numbers in the second column are added, so giving us,

Step 4

Then recently written number 7 is multiplied by the known root, – 2,

As 14 comes as a result, you need to write it down on the second row over the line,

Step 5

Like previously the numbers in this column added, (14 – 2 = 12)

Step 6

And you need to go on with the process,                                                                  

 

Step 7

When you have zero at the bottom row, it gives the confirmation that x = – 2 is a root of the original cubic. At this stage, you got the first three numbers in the bottom row as the coefficients in the quadratic,

x2– 7x+12

Hence, you reduced your cubic to,

(x+2)(x2 – 7x + 12) =0

Step 8

After applying the quadratic term, the equation comes like this,

(x +2) (x – 3) (x – 4) = 0

Resulting, you get the solution as x = -2 or or 4.

ANOTHER EXAMPLE:

The equation is,

x3– 7x-6=0

Step 1

You can simply try x = 1, after putting the value of x, you will get,

(-1)3 – 7(-1) -6

The result comes as zero, so it is proved that x +1 is a factor and the cubic can be written in the form,

(x+1)(x2+ax+b)= 0

Step 2

After applying the synthetic division, like above example, you will take the coefficients of the original cubic equation, which are 1, 0, -7 and -6, you need to write down the know root x = -1 to the right of the vertical line, giving us,

Step 3

Multiply the brought down number 1 by the known root x = -1, and put down the result (-1) at the second row, like this,  

Step 4

The numbers of the second column are added to the first column, giving us,

And continue the process adding numbers in this column, until you find ‘0’ as a result,

Step 5

As you add more numbers to the second column by following the synthetic division process, you will come up with,

When you come up with zero, it is confirmation that x = -1,

Hence, you get the coefficients in the quadratic as the first three numbers in the bottom row, so the quadratic is, x-x – 6

Step 6

Hence, the cubic reduced to quadratic,

(x+1)(x2-x- 6) =0

The factorized result is,

(x +1)(x – 3)(x + 2) = 0

The three solutions or roots to the cubic equation are x = -2, -1 or 3

Are you looking for Assignment Help with 100% accuracy, My assignment help .com have 3000+ Professional Writers are here to help with your Assignment Writing & also provide 24×7 Online Support.

How MyAssignmenthelp.com can help students in maths assignment

MyAssignmenthelp.com has carefully evaluated and assembled a highly-skilled team of in-house experts who are capable of giving the finest maths homework help you deserve. Your paper will be written by experienced writers in a professional manner. Assignment experts make sure that there are no traces of plagiarism in your content, they only deliver genuine work within the given time limit. If you think sample assignments can help you in your assignment, MyAssignmenthelp.com offers free samples. Our customer support agents can guide you on how to get free assignments quotes and place your orders.

Related Blog:

Glossary:

  • Real Root : A real number x will be called a solution or a root if it satisfies the equation.
  • Polynomials: An expression consisting of variables, constants and operator functions like (-) (+) (*) (/). Ex: x2 - 4x +7 
  • Variable: Any quantity that changes in a mathematical equation is called a variable. Commonly, we use an alphabet to represent a variable. x is the variable in the above equation.
  • Factor theorem: It is a theorem which links factors and zeros of polynomials.

Related Posts

Save Time & Excel Academics

Share your requirements & connect with top tutor !.

watch
question
We will use e-mail only for:

arrow Communication regarding your orders

arrow To send you invoices, and other billing info

arrow To provide you with information of offers and other benefits

loader
Add File

Files Missing!

Please upload all relevant files for quick & complete assistance.

2,019,747

Orders

4.9/5

Overall Rating

5,178

Experts

Still in Two Minds? The Proof is in Numbers!

Genuine Reviews With a Rating of 4.9/5.

student rating student rating student rating student rating student rating
20 May 2022

HRM

Home Work: 9 Pages, Deadline: 4 days

Got the great grading, very surprise and happy. The expert work hard and understanding the questions requirements, so got the great grading, thank you

flag User ID: 4***0 Central District, Hong Kong

student rating student rating student rating student rating student rating
20 May 2022

Statistics

Programming: 1 Page, Deadline: 18 hours

They are very professional and they care about you grades. The work done was so nice and meet all the requirement of the assesment..thanks team

flag User ID: 9***96 Sydney, Australia

student rating student rating student rating student rating student rating
20 May 2022

Healthcare

Assignment: 3 Pages, Deadline: 14 days

Excellent work from an expert, though I have to add something from my side. It was received on time as well, though I didn\'t get the marks yet hope ...

flag User ID: 4***47 Sydney, Australia

student rating student rating student rating student rating student rating
20 May 2022

Criminal Law

Home Work: 4 Pages, Deadline: 1 day

It was quick and very well written. I was very happy with the work and they used very deep and vast knowledge.

flag User ID: 9***93 London, Great Britain

student rating student rating student rating student rating student rating
20 May 2022

Healthcare

Home Work: 1 Page, Deadline: 1 day

Thanks for the solution and the assignment was well written got what was asked for.

flag User ID: 4***47 Sydney, Australia

student rating student rating student rating student rating student rating
20 May 2022

Accounting

Home Work: 3.4 Pages, Deadline: 2 days

The work was done pretty good by the expert and i got good grades. They wrote every details very well.

flag User ID: 2***21 Kenthurst, Australia

student rating student rating student rating student rating student rating
20 May 2022

Medical

Essay: 1 Page, Deadline: 7 days

Still waiting on results but overall happy with the assessment, I hope I get a good mark

flag User ID: 4***37 Sydney, Australia

student rating student rating student rating student rating student rating
20 May 2022

Sociology

Essay: 2 Pages, Deadline: 3 days

Informative with Excellent written Research Assignment. Will now and in the future definitely recommend myself this phenomenal website. Thank you!

flag User ID: 9***78 Plymouth, United States

student rating student rating student rating student rating student rating
20 May 2022

Chemistry

Programming: 1.7 Pages, Deadline: 4 days

I really wanted to let you guys no the quality of work is great. I really appreciate the excellent service and superior detail you\'ll did. Thank yo ...

flag User ID: 8***24 United States

student rating student rating student rating student rating student rating
20 May 2022

Economics

Home Work: 1 Page, Deadline: 15 hours

Excellent service and assistance. I appreciate the quality of work and quick turnaround

flag User ID: 8***24 United States

student rating student rating student rating student rating student rating
20 May 2022

Finance

Assignment: 8 Pages, Deadline: 16 days

Very good writing. All the points were covered and was well written. Will definitely recommend to others

flag User ID: 1***20 Daceyville, Australia

student rating student rating student rating student rating student rating
20 May 2022

Geography

Essay: 1.1 Pages, Deadline: 3 days

Great work with fast service. It was well detailed according to what I was looking for.

flag User ID: 8***24 United States

student rating student rating student rating student rating student rating
20 May 2022

Management

Thesis: 6 Pages, Deadline: 15 hours

I am totally satisfied with your assignment and I am looking forward to working with you thank you.

flag User ID: 9***57 Australia

student rating student rating student rating student rating student rating
19 May 2022

Project Management

Home Work: 7.4 Pages, Deadline: 1 day

it was really the best assignment done, the tutor wad so helpful and very fast answering my request

flag User ID: 5***9 Saint Kilda, Australia

student rating student rating student rating student rating student rating
19 May 2022

Sociology

Programming: 6 Pages, Deadline: 11 days

very impressed with the quality of work and grades thank you and the duration of time it took to be done and all the specifications were met according ...

flag User ID: 6***23 Nicosia, Cyprus

student rating student rating student rating student rating student rating
19 May 2022

Mechanical Engineering

Assignment: 2 Pages, Deadline: 6 days

I got full marks on this assignment, thank you very much to the expert. I really appreciate it.

flag User ID: 5***55 United States

student rating student rating student rating student rating student rating
19 May 2022

Finance

Programming: 4.8 Pages, Deadline: 10 hours

The work was okay. I have sent it to my student. He is checking it and if there will be any updates I will let you know

flag User ID: 1***79 Dublin, Ireland

student rating student rating student rating student rating student rating
19 May 2022

Programing

Programming: 3 Pages, Deadline: 1 day

The work was okay. I have sent it to my student. He is checking it and if there will be any updates I will let you know. Thank you so much

flag User ID: 1***79 Dublin, Ireland

student rating student rating student rating student rating student rating
19 May 2022

Programing

Programming: 2 Pages, Deadline: 2 days

Till now what I saw was okay. My student is verifying and checking the project. Any update will be there I will let you know.

flag User ID: 1***79 Dublin, Ireland

student rating student rating student rating student rating student rating
19 May 2022

Ancient History

Essay: 1.2 Pages, Deadline: 2 days

this homework was for my history class and it was good I did get a good grade for it thank you.

flag User ID: 6***66 United States

5% Cashback

On APP - grab it while it lasts!

Download app now (or) Scan the QR code

*Offer eligible for first 3 orders ordered through app!

screener
ribbon
Have any Query?