Solving a cubic equation online

In this article, we will cover how to solve a Cubic Equation in 3 ways. We also provide a program for instant calculation of the solutions to any cubic equation where the user can enter its parameters a, b, c, and d. Additionally, we’ll provide several cubic equation exercises to help you understand.

The general form of a cubic equation:

Where a, b, c are the coefficients of x, while d is a numerical constant.

Equation conditions: a is not equal to zero.

Method for Solving a Cubic Equation

Students usually solve a cubic equation by converting it into a quadratic equation, either by simplifying and eliminating some values in one way or another, or by grouping the equation into parentheses (containing x values of the first or second degree) multiplied by each other, where the other side of the equation equals zero. Therefore, each of those parentheses equals zero and holds one or more values for the equation’s solutions.

Solving a Cubic Equation by Factoring

We can solve a cubic equation by factoring. Suppose we have the following equation:

2x³ + 3x² – 11x – 6 = 0

Note that the coefficient d=6 has prime factors 1, 2, 3, 6 (the factors of a number are the numbers you can multiply together to get the number. See Prime Numbers). Let’s substitute the prime factors into the equation:

f (1) = 2(1)³ + 3(1)² – 11×(1) – 6 ≠ 0 The equation is not satisfied
f (–1) = –2 + 3 + 11 – 6 ≠ 0 The equation is not satisfied
f (2) = 16 + 12 – 22 – 6 = 0 Satisfied

So, without a doubt, the first solution to the equation is 2

By factoring the equation into parentheses (Synthetic division), we can find the remaining solutions. Assuming 2 is one of the solutions, we write the equation according to the formula:

= (x – 2) (ax² + bx + c)

Yes, we can write it in the previous form, where the equation resolves into two parentheses: one containing a solution (x - solution) multiplied by a parenthesis containing the formula for a quadratic equation. Now let’s expand the parentheses.

= ax³-2ax²+bx²-2bx+cx-2c

= ax³+(-2a+b)x²+(-2+c)bx-2c

By balancing with the original equation, we find that:

-2c = -6
c=3

a=2

-2a+b=3
-2*2+b = 3
b=7

So the resulting equation is:

(x – 2) (2x² + 7x + 3) = 0
(x – 2) (2x + 1) (x +3) = 0

By setting each term equal to zero, we find that the three roots are 2, -0.5, -3

Solving a Cubic Equation by Euclidean Division

Let’s assume the following cubic equation:

x³ – 2x² – 5x + 6 = 0

The factors for this equation, according to the value d = 6, are 1, 2, 3, and 6. Let’s try dividing by (x-2)

(x³ – 2x² – 5x + 6) ÷ (x-2)

Performing the division process: In the previous figure, we wrote the coefficients of the original equation in the first row.

The root is 2, so we divide by 2 as follows:

1. The number 1 comes down as it is.
2. Multiply the resulting 1 by 2. The result is 2, which we place under the coefficient -2.
3. Add -2 with 2. The result is 0.
4. Multiply 0*2. The result is 0, which we place under the coefficient -5.
5. Add -5 with 0. The result is -5.
6. Multiply -5 * 2. The result is -10.
7. Place -10 under the coefficient 6.
8. Add -10 with 6. The result is -4.

If the last number is not zero, this means that 2 is not a solution to the equation, so we should try another factor!!

It would have been better to test the factor as a solution to the equation by substituting it directly into the equation! By substituting into the equation, we conclude that 3 is one of the solutions. Now let’s try the division process.

The last division result is zero, which means that the root 3 is correct and is a solution to the equation. But what about the other resulting numbers? What do they mean?

The remaining numbers are the coefficients of the quadratic equation that contains the rest of the roots:

ax² + bx + c = 0

a, b, and c are respectively:

a = 1 , b = 1, c = -2

The equation is:

x² + x -2 = 0

The solution to this equation is simple, and the other roots for the cubic equation are 1 and -2

Cardano’s Method

Have you wondered how we solved the equation in the calculator program above? We certainly didn’t rely on factoring, but rather on Cardano’s method. Cardano’s method for solving a cubic equation is similar to the traditional solution for a quadratic equation using Delta, but it is certainly more complex. However, it is comprehensive, no matter how difficult the solution. Please comment if you are interested in the article, as our writers are preparing to write the method in a simplified style. In the meantime, we can guide you to this video on solving an equation using Cardano’s method.

https://www.youtube.com/watch?v=xgSz9mp2mxU Solving a cubic equation using Cardano’s method