2 Easy Methods for Solving Cubic Equations
A cubic equation is in the form of ax3 + bx2 + cx + d = 0 and the highest exponent in these equations is 3. There are numerous ways for solving a cubic equation and two most common and effective methods for solving a cubic equation are illustrated below along with steps. The first method is for equations with d=0 while for the equations with non-zero d value integer solutions method is adopted. The figure below illustrates the general form of a cubic equation.
1. Solving Cubic Equations without a Constant
There are five simple and easy steps of solving a cubic equation without a constant. All four steps are illustrated below:
1. Check Constant Value in the Equation
In order to use the following method for solving a cubic equation, it is important to identify whether the equation contains a constant value or not. If a constant value is present, you cannot use the following method.
2. Factor an x Out
The next step is factoring out an ‘x’ since all the terms in the equation contains an ‘x’. For instance, your equation will become like this: x (ax2 + bx + c). The figure below illustrates an example of factoring out an x in a cubic equation.
3. Determining the Roots with Factorization
In the following method, one root is 0 while with the help of factorization, the rest of 2 roots could be determined. The figure below illustrates the use of factorization to determine the roots of the cubic equation.
4. Determining Roots with the Help of Quadratic Equation
If you are unable to find the roots manually, then, another effective method is the use of the quadratic equation. The figure below illustrates the use of a quadratic equation where values of a, b, and c are inserted in the formula to determine the roots.
In the following method, one root is always 0 because x is directly equated with 0 while the rest two roots are determined with either factorisation or quadratic equation.
2. Finding Integer Solutions with Factor Lists
Another effective method of solving a cubic equation is the use of integer solution method, and there are five steps that are illustrated.
1. Ensuring a non-constant Value
The first step is to ensure that the cubic equation has a non-zero d value and an illustration is shown in the figure below.
As shown in the above figure, the following cubic equation has non-zero d value; therefore, in this case, the integer solution method could be adopted.
2. Find Factors of a and d
The next step is to determine the factors of a, and d in the cubic equation.
The above figure shows the factors of a, and d of the cubic equation. In the following equation a=2 and d=6 where factors of a are 1, and 2 whereas the factors of d are 1, 2, and 3.
3. Dividing Factors of a by d
The third step is the dividing of factors of a with the factors of d and the figure below shows the dividing of factors of a by d.
As shown in the above figure few whole numbers while various fractions are obtained after the dividing of factors of a by d. In addition to this, integer solutions of the cubic equation will be either whole number or negative number in the list.
4. Plug in the Integers Approach
After you have obtained a list of possible solutions, the next step is to put each integer one by one to get 0. It is generally an easy process, but it could be time-consuming. The figure below shows the illustration of putting different integers such as 1 and -1 in place of x in order to obtain a 0 in the equation.
As in the following cubic equation after putting x=-1, the equation becomes 0; hence, the root of this equation was found to be -1. The process of putting integers is continued until the equation becomes equal to zero.
5. Synthetic Division Approach
There is another effective approach that is the use of synthetic division which is quite complex as compared to plug in integer approach but is a more effective and quick method for finding roots of a cubic equation. In the following method, integer values obtained are synthetically divided with the coefficients of a, b, c, and d of the cubic equation. If 0 remainder is obtained, it means that the value is the answer to the cubic equation.
The above figure shows the synthetic division with integer -1 and as 0 remainders are obtained; hence, -1 is the answer of the cubic equation.
These are the two most effective and common methods that help in solving all kind of cubic equations and determining the roots of the cubic equation.
Mark J. Pattinson is a mathematician with a PhD degree in the same domain. He has associated with “MyAssignmentHelp UK” since the last eight months and writes regularly. He is a valuable addition to the writing team due to versatile writing style.