Cubic Equation Calculator + Online Solver With Free Steps

A Cubic Equation Calculator is used to find the roots of a cubic equation where a Cubic Equation is defined as an algebraic equation with a degree of three.

An equation of this type has at least one and at most three real roots, and two of them can be imaginary.

This calculator is one of the most sought-after calculators in the field of mathematics. This is because solving a cubic equation by hand is not opted for usually. The input boxes are set up to provide simplicity and total efficiency for the entry of problems and getting results.

What Is a Cubic Equation Calculator?

The Cubic Equation Calculator is a calculator that you can use in your browser to solve for roots of Cubic Equations.

Figure-1 Cubic Equation Calculator

This is an online calculator which you can use at any place and time. It doesn’t require anything other than a problem to solve from you. You don’t need to install or download anything to use it.

You can simply enter the coefficients of your variables in the input boxes on your browser and get your desired results. This calculator can solve third-degree polynomials using algebraic manipulations and operations.

How To Use a Cubic Equation Calculator?

You can use Cubic Equations Calculator by entering the values of coefficients of each variable of a cubic equation in the specified fields.

It is a very convenient tool for finding solutions to your algebraic problems, and here’s how to use it. You first need to have a cubic equation which you wish to get the roots for. Once you have a problem in need of a solution, you can follow the given steps to acquire the best results.

Step 1

Start by placing the coefficients of each variable in the cubic equation inside their respective input boxes. There are four input boxes: a, b, c, and d, each representing the overall cubic equation: $ax^3+bx^2+cx+d = 0$.

Step 2

Once all the values are placed in the input boxes, all you are left with is to press the Submit button, after which the result of your problem is expressed in a new window.

Step 3

Finally, if you want to keep using the calculator, you can update the inputs inside the new window and get new results.

How Does the Cubic Equation Calculator Work?

The Cubic Calculator works by calculating the algebraic solution to the polynomial with the degree three. Such an equation can have the following form:

\[ ax^3 + bx^2 + cx + d = 0\]

To solve a Third-Degree Polynomial, you need to first consider the type of the polynomial. If the polynomial doesn’t have a constant term attached to it, then it becomes very easy to solve, but if your polynomial has a constant term within it, then it has to be solved using a set of other techniques.

For Cubic Equations Without the Constant Term

A Cubic Equation which does not have a constant term in it allows one to break it up into a product of a quadratic and a linear equation.

It is a renowned fact that linear equations can make up any degree of the polynomial, based on the multiplicative properties of a polynomial. A cubic equation of the form, $ax^3+bx^2+cx = 0$ is the one referred to as an equation without the constant term.

This type of cubic equation can be simplified into their respective quadratic and linear equations i.e., $x(ax^2+bx+c) = 0$ by using algebraic manipulations.

Once you have a product of quadratic and linear equations acquired, you can carry it forward by equating it to zero. Solving for x will give the results, given we have ways of solving linear as well as quadratic equations where the methods to solve quadratic equations are Quadratic Formula, Completing Squares Method, etc.

For Cubic Equations With the Constant Term

For a Cubic Polynomial containing a constant term, the above method loses doesn’t help. Because of this, we rely on the fact that the roots of an algebraic equation are supposed to equate the polynomial to zero.

So Factorization is one of the many ways to solve this type of algebraic problem.

Factorization of any degree of polynomial begins the same way. You start by taking integers on the number line and place x, the variable under question equal to those values. Once you find 3 values of x, you have the solution roots.

An important phenomenon to observe is that the degree of the polynomial represents the number of roots it will produce.

Another solution to this problem would be Synthetic Divisions, which is a more trustworthy quick approach and can be very challenging.

Solved Examples

Example 1

Figure-2 Cubic Equation Solved Example

Consider the following cubic equation, $1x^3+4x^2-8x+7 = 0$, and solve for its roots.

Solution

Starting with the entry of the a, b, c, and d corresponding with the respective coefficients of the cubic equation in question.

The real root of the equation is eventually given as:

\[x_1 = \frac{1}{3} \bigg(-4-8\times5^{\frac{2}{3}}\sqrt[3]{\frac{2}{121-3\sqrt{489}}} – \sqrt[3]{\frac{5}{2}(121-3\sqrt{489}}\bigg) \approx 5.6389\]

Whereas the complex roots are found to be:

\[x_2 \approx 0.81944 – 0.75492i, x_3 \approx 0.81944 + 0.75492i\]

Figure-3 Graph Of the Above Example

Example 2

Consider the following cubic equation, $4x^3+1x^2-3x+5 = 0$, and solve for its roots.

Solution

Starting with the entry of the a, b, c, and d corresponding with the respective coefficients of the cubic equation in question.

The real root of the equation is eventually given as:

\[x_1 = \frac{1}{12} \bigg(-1 – \frac{37}{\sqrt[3]{1135-6\sqrt{34377}}} – \sqrt[3]{1135 – 6\sqrt{34377}}\bigg) \approx -1.4103\]

Whereas the complex roots are found to be:

\[x_2 \approx 0.58014 – 0.74147i, x_3 \approx 0.58014 + 0.74147i\]

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