Cubic equation

Do you want to solve cubic equation online? Below you will find a calculator to solve them immediately. Just type in the value of the coefficients of the equation, press the calculate button and you're done.

If you still have any doubts, in this article we will teach you how to solve equations of degree threeexercises and many interesting resources.

 




How do you solve a cubic equation?

For learning to solve a cubic equation it is important that we know that they are of this form:

Ax3 + Bx2 + Cx + D = 0

That is what a complete cubic equation looks like and that is what we will learn to solve next using several methods. It is important for you to know that this type of equations always have three solutions or roots. Furthermore, depending on the value of the independent term D, the solutions can be as follows:

  • D > 0One of the solutions is real and the other two are complex.
  • D = 0All solutions are real and at least two are equal.
  • D < 0all solutions are real and completely different.

Now that we know the basics of degree 3 equations, let's see how to solve them. If you want to learn how to solve incomplete third degree equations, read on to find out how to do it.

By Ruffini

 


Using Ruffini's rule is one of the most recommended methods for solving cubic equation step by step.

The process is much easier to understand if we show it to you in video than if we write it down, so we leave you with one of the best ones we have found on YouTube. Don't worry if instead of Ruffini you hear the professor talking about synthetic division, it's the same thing.

Without Ruffini

In the case that Ruffini does not convince you to calculate cubic equations, you can use this other method based on the calculation of various formulas and values for this equation of degree three:

x3 + a1x2 + a2x + a3 = 0

The first step is to find the following values:

Third degree formulas

With the values of Q, R, S1 and S2 already computed, now we substitute in the following general equations to find the three solutions of the cubic equation:

General equations third degree

General formula

If none of the above methods convinces you, here is what you can do with three formulas for each of the solutions of the equationbut I warn you that they are quite tedious to solve.

Formula for the first solution X1:

First Solution for third degree formula

Formula for the second solution X2:

Second Solution for third degree formula

Formula for the third solution X3:

Third Solution for third degree formula

We hope that at least one of the three methods that we have proposed to you for solving cubic equation is useful for you. In addition, you can always check if the result you have obtained is correct by using our calculator.

Incomplete cubic equation

Sometimes they will send us solving incomplete cubic equation or without an independent term that looks like this:

Ax3 + Bx2 + Cx = 0

In these cases, one of the solutions will always be X = 0. The other two will be calculated by taking out the common factor in this way:

x(AX2 + Bx + C) = 0

If we look at it, the other two missing solutions of the incomplete cubic equation are found in the second-degree equation between the parentheses that we have to solve. If you do not know how to do it, you can use our second degree equations calculator.

Cubic equation solved

Are you looking for exercises solved cubic equation? Here are some examples with the solutions so that you can try to solve them yourself:

First exercise: x3 - 6x2 + 3x + 10 = 0

Solutions:

  • x1 = 5
  • x2 = 2
  • x3 = -1

Second exercise: 2x3 - 5x2 + 3x = 0

Solutions:

  • x1 = 0
  • x2 =1
  • x3 = 3/2

How to solve cubic equation on the calculator

Third degree equations calculator

If you have a relatively modern scientific calculator, you can solve cubic equation with it. To see how it is done, we will use one of the brand name Casio as they are the most popular among high school, college or university students.

To calculate all the solutions of a cubic equation with the calculator you have to look for the "MODE SETUP" or "MENU CONFIG" button. which is usually located in the upper right corner. Click on it and from all the options that appear on the screen, select the one represented as "EQN".

If everything has gone well, the screen will display equations of different degrees. We will stay with the one of degree 3 that looks like this:

aX3 + bX2 + cX + d = 0

We select it and the next step will consist of write down the values of coefficients a, b, c and d. When you have typed them, press the equals key (=) to display the solutions.

You will probably have to use the four-way pad (in the picture above you can see it) to display all the solutions of the equation you have written.

How the cubic equation calculator works

Online third degree equations calculator

In case it is not clear to you how our cubic equation calculator online, we are going to explain it in more detail.

As we have already seen, the cubic equation must have the following format:

Ax3 + Bx2 + Cx + D = 0

You simply have to type in the tool the values of A, B, C and D. If any of these values is 0, you must type it in. For example, if the cubic equation you want to solve online is 2X3  - 4X2  - 22X + 24 = 0, then the values to write in the calculator are as follows:

  • A = 2
  • B = -4
  • C = -22
  • D = 24

Then just press the calculate button and you will get the result immediately.

If you have any doubts about how to solve a cubic equation, leave us a comment and we will help you as soon as possible.

2 thoughts on “Ecuaciones de tercer grado”

    • Hello Ruben,

      The i you mention refers to the imaginary unit. Note that depending on the value of the discriminant, the third degree equation may have 3 real roots or one may be real and two may be complex (not real) roots expressed with the 'i'.

      Greetings!

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