Cubic Equation Solver

Introduction to Cubic Equation Solver:

Cubic equation solver is an online tool that helps you to find third-degree polynomial equation roots.

Our cubic equation calculator evaluates the three-variable equation that gives three real roots or at least one real root and two imaginary roots in a couple of minutes.

What is the Cubic Equation?

Cubic equation is a polynomial equation whose highest degree is three that has at least one variable as a real number or two may be imaginary numbers.

It is also possible that a cubic equation may give three possible real number roots without any imaginary number. Cubic equations can be solved by factorization or synthetic division methods.

Formula Behind Cubic Equation Calculator:

Cubic equation formula of the polynomial function has a maximum degree is three. The Cubic equation solver uses the same formula for evaluation which is,

$$ ax^3 + bx^2 + cx + d \;=\; 0 $$

Where,

a,b,c,d are coefficients of polynomials and a is not equal to zero. Otherwise cubic equation changes into a quadratic equation.

Steps by Step Evaluation of Cubic Solver:

The cubic formula calculator uses the simplest method to give a solution of the cubic equation of polynomials so that you can easily grasp the core of this concept. You just need to enter the specific function and you will get a solution in one click due to its up-to-date algorithm.

When you enter the given cubic function in the cubic function calculator, it will analyze the given polynomial function.

Then cubic polynomial calculator will decide which method is best for solving the third degree of the polynomial equation after identification. These methods are factorization and synthesis division methods for cubic equations.

Factorization Method:

Factorization is one of the common factors in finding out the root of x from cubic equations (ax³+bx²+cx+d) using the cubic equation calculator. However, all cubic equations are not solved by the factorization method.

At first Cubic equation solver finds the common prime factor (x-a) from the given cubic equation when you add different numbers in p(x) and it satisfies the equation as p(x) = 0.

After finding that root we find (x-a) and the rest of the equation becomes x polynomial quadratic equation (bx²+cx+d).

Now quadratic equations are easily solved with the factorization method or quadratic equation formula to give the root of x. You can get a better understanding of this concept in the below example for more clarity.

Synthetic Division Method:

In the synthesis division method, two or three roots are selected and then the cubic formula solver divides this root with all the coefficients in the equation one by one so that at last it gives zero answers. The root that is equal to zero is our particular root and the other roots are useless.

Now the cubic solver finds its first root which is (x-a) and now the equation becomes a polynomial quadratic equation (x-a) (bx²+cx+d). This quadratic equation is now solved with the factorization method easily.

You can get an elaborate method practically in the given example below and will understand it better.

Solved Example of the Cubic Equation:

An example with the step-by-step solution of the cubic equation is given to understand the results given by the Cubic equation solver and to know manual calculations so solve such problems.

Example 1:

$$ x^3 + 4x^2 + 5x + 2 $$

Solution:

Let,

$$ p(x) \;=\; x^3 + 4x^2 + 5x + 2 $$

\begin{array}{c|rrrr}& 1 & 4 & 5 & 2 \\ & 0 & 1 & 5 & 10\\ {\color{red}1} \\ \hline & 1 & 5 & 10 & |\phantom{-} {\color{yellow}12} \end{array}

\

\begin{array}{c|rrrr}& 1 & 4 & 5 & 2 \\ & 0 & -1 & -3 & -2 \\ {\color{red}1} \\ \hline & 1 & 3 & 2 & |\phantom{-} {\color{yellow}0} \end{array}

So we get,

$$ 12 \neq 0 $$

As we get 0 as remainder. So, -1 is one of the solutions. Create a quadratic polynomial by factoring this, you will get two factors,

$$ x^2 + 3x + 2 \;=\; (x+1)(x+2) $$

$$ x^3 + 4x^2 + 5x + 2 \;=\; 0 $$

$$ (x+1)(x+1)(x+2) \;=\; 0 $$

Equating each factor to zero,

$$ x + 1 \;=\; 0, x + 1 \;=\; 0 \;and\; x + 2 \;=\; 0 $$

$$ x \;=\; -1, -1 , -2 $$

Example 2:

Calculate the roots of the cubic equation of the following:

$$ 2x^3 + 3x^2 - 11x - 6 \;=\; 0 $$

Solution:

The possible factors are 1, 2, 3, and 6 as d = 6, applying the factor theorem by trial and error,

$$ f(1) \;=\; 2 + 3 - 11 - 6 \neq 0 $$

$$ f(-1) \;=\; -2 + 3 + 11 - 6 \neq 0 $$

$$ f(2) \;=\; 16 + 12 - 22 - 6 \;=\; 0 $$

x = 2 is the first root, so other roots can be determined by the synthetic division method,

$$ \;=\; (x - 2)(ax^2 + bx + c) $$

$$ \;=\; (x - 2)(2x^2 + bx + 3) $$

$$ \;=\; (x -2)(2x^2 + 7x + 3) $$

$$ \;=\; (x -2)(2x + 1)(x +3) $$

Therefore, the solutions are,

$$ x \;=\; 2, x \;=\; - \frac{1}{2} \;and\; x \;=\; -3 $$

Evaluation in the Cubic Equation Solver:

Cubic equation calculator has a user-friendly interface so that you can use it to calculate the polynomial differential equation in less than a minute.

Before adding the input value to the cubic polynomial solver, you must follow some simple steps so that you do not experience trouble during the calculation process. These steps are:

1. Enter the coefficient value of a,b,c, and d in the input box of cubic function solver.
2. Review your polynomial function before hitting the calculate button to start the calculation process.
3. Click the “Calculate” button of cubic solver to get the desired result of your given cubic equation.
4. If you want to try out our third degree polynomial calculator first then you can use the load example that gives you better clarity about its working.
5. Click on the “Recalculate” button to get a new page for solving more cubic equation problems.

Outcome from Cubic Formula Calculator:

Cubic equation solver gives you the solution of the given numerical problem when you add the input in it. It provides you with solutions in a step-wise process in less than a minute. It may contain the following:

• Result option gives you a solution for the cubic equation
• Possible step option provides you solution with all the calculation steps of the cubic equation problem

Advantages of Cubic Function Calculator:

Cubic solver will give you tons of advantages whenever you use it to calculate polynomial third-degree problems. These advantages are:

• Our cubic formula calculator saves your time and effort from doing lengthy calculations of the cubic equation problem.
• Our cubic formula solver is a free-of-cost tool so you can use it to find the polynomial cubic equation.
• The cubic equation calculator is a versatile tool that allows you to solve various types of coefficients of polynomial functions.
• You can use this cubic polynomial calculator for practice so that you get a strong hold on the cubic equation problems.
• It is a reliable tool that provides you with accurate solutions every time whenever you use it to calculate the given cubic equation problem.
• The cubic equation solver provides a solution with a complete process in a step-wise process so that you get a better understanding of this cubic polynomial equation concept.