BrightChamps Logo
Login
Creative Math Ideas Image
Live Math Learners Count Icon101 Learners

Last updated on August 5th, 2025

Math Whiteboard Illustration

Math Formula for Cubic Equations

Professor Greenline Explaining Math Concepts

In algebra, a cubic equation is a polynomial equation of degree three. It is expressed in the form ax^3 + bx^2 + cx + d = 0. In this topic, we will learn the formulas and methods to solve cubic equations.

Math Formula for Cubic Equations for Filipino Students
Professor Greenline from BrightChamps

List of Math Formulas for Solving Cubic Equations

Cubic equations involve polynomials of degree three. Let’s learn the formulas to solve cubic equations using different methods.

Professor Greenline from BrightChamps

General Form of a Cubic Equation

The general form of a cubic equation is expressed as:

 

ax3 + bx2 + cx + d = 0, where a, b, c, and d are constants and a ≠ 0.

Professor Greenline from BrightChamps

Solving Cubic Equations using Factorization

One method to solve cubic equations is by factorization. This involves finding a root of the equation and dividing the cubic equation by (x - root) to simplify it into a quadratic equation, which can then be solved using standard methods.

Professor Greenline from BrightChamps

Cardano’s Method for Solving Cubic Equations

Cardano’s method is used for solving general cubic equations. It involves a series of substitutions and transformations to reduce the cubic equation to a depressed cubic form and solve it. The process can be complex and is typically used when other methods are unsuitable.

Professor Greenline from BrightChamps

Importance of Cubic Equation Formulas

Cubic equations appear in various fields, including physics, engineering, and computer graphics. Understanding how to solve them is crucial for modeling and solving real-world problems involving cubic relationships.

Professor Greenline from BrightChamps

Tips and Tricks to Solve Cubic Equations

Students often find solving cubic equations challenging. Here are some tips and tricks to master them:

 

- Start by checking for any obvious roots, such as x = 0 or simple integer values.

 

- Use the Rational Root Theorem to identify potential rational roots.

 

- Practice using synthetic division to simplify the equations quickly.

Max Pointing Out Common Math Mistakes

Common Mistakes and How to Avoid Them While Solving Cubic Equations

Students often make errors when solving cubic equations. Here are some mistakes and ways to avoid them to master these equations.

Mistake 1

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Ignoring Potential Rational Roots

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students sometimes overlook the Rational Root Theorem, which can help identify potential roots. To avoid this, always test simple rational roots before proceeding with more complex methods.

Mistake 2

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Errors in Synthetic Division

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Errors can occur when using synthetic division to simplify cubic equations. To avoid these, double-check each step of the division process and verify the results.

Mistake 3

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Misapplying Cardano’s Method

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students may misapply Cardano’s method due to its complexity. To avoid this, thoroughly understand each step and practice with simpler examples before tackling more complex equations.

Mistake 4

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Confusing Terms and Coefficients

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students often confuse the terms and coefficients in the equation, leading to calculation errors. To avoid this, carefully identify each coefficient and term before starting the solution process.

Mistake 5

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Overlooking Factorization

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Sometimes students fail to recognize when a cubic equation can be factored easily. Always attempt factorization first, as it can significantly simplify the solving process.

arrow-right
Max from BrightChamps Saying "Hey"
Hey!

Examples of Problems Using Cubic Equation Formulas

Ray, the Character from BrightChamps Explaining Math Concepts
Max, the Girl Character from BrightChamps

Problem 1

Solve the cubic equation x^3 - 6x^2 + 11x - 6 = 0?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

The roots are x = 1, x = 2, and x = 3

Explanation

First, test for rational roots using the Rational Root Theorem.

The potential roots are ±1, ±2, ±3, ±6.

Testing x = 1, we find it is a root.

Divide the cubic polynomial by (x - 1) to get a quadratic: x^2 - 5x + 6.

Factor the quadratic: (x - 2)(x - 3).

Thus, the roots are x = 1, x = 2, and x = 3.

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 2

Solve the cubic equation 2x^3 - 4x^2 - 22x + 24 = 0?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

The roots are x = -2, x = 2, and x = 3

Explanation

First, check for rational roots using the Rational Root Theorem.

The potential roots are ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±24.

Testing x = 2, we find it is a root.

Divide the cubic polynomial by (x - 2) to get: 2x^2 - 11x + 12.

Factor the quadratic: (2x - 3)(x - 4).

Thus, the roots are x = -2, x = 2, and x = 3.

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 3

Find the roots of the cubic equation x^3 + 3x^2 - 4x - 12 = 0?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

The roots are x = -3, x = -2, and x = 2

Explanation

Using the Rational Root Theorem, test potential roots: ±1, ±2, ±3, ±4, ±6, ±12.

Testing x = -2, we find it is a root.

Divide the cubic polynomial by (x + 2) to get: x^2 + x - 6.

Factor the quadratic: (x - 2)(x + 3).

Thus, the roots are x = -3, x = -2, and x = 2.

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 4

Solve the cubic equation x^3 - 3x^2 - 4x + 12 = 0?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

The roots are x = -2, x = 2, and x = 3

Explanation

Using the Rational Root Theorem, test potential roots: ±1, ±2, ±3, ±4, ±6, ±12.

Testing x = 3, we find it is a root.

Divide the cubic polynomial by (x - 3) to get: x^2 + 4x - 4.

Factor the quadratic: (x - 2)(x + 2).

Thus, the roots are x = -2, x = 2, and x = 3.

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 5

Find the roots of the cubic equation x^3 - x^2 - 9x + 9 = 0?

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

The roots are x = -3, x = 1, and x = 3

Explanation

Using the Rational Root Theorem, test potential roots: ±1, ±3, ±9.

Testing x = 3, we find it is a root.

Divide the cubic polynomial by (x - 3) to get: x^2 + 2x - 3.

Factor the quadratic: (x + 3)(x - 1).

Thus, the roots are x = -3, x = 1, and x = 3.

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Ray Thinking Deeply About Math Problems

FAQs on Cubic Equation Formulas

1.What is the general form of a cubic equation?

The general form of a cubic equation is ax^3 + bx^2 + cx + d = 0, where a, b, c, and d are constants and a ≠ 0.

Math FAQ Answers Dropdown Arrow

2.What is the Rational Root Theorem?

The Rational Root Theorem states that any rational root of the polynomial equation ax^n + bx^(n-1) + ... + k = 0 is of the form ±p/q, where p is a factor of the constant term k and q is a factor of the leading coefficient a.

Math FAQ Answers Dropdown Arrow

3.How does Cardano’s method work?

Cardano’s method involves transforming the cubic equation into a depressed cubic form and solving it through a series of substitutions and algebraic manipulations.

Math FAQ Answers Dropdown Arrow

4.What are common mistakes when solving cubic equations?

Common mistakes include overlooking potential rational roots, errors in synthetic division, and confusing terms and coefficients.

Math FAQ Answers Dropdown Arrow

5.How can cubic equations be applied in real life?

Cubic equations are used in physics to calculate volumes, in engineering for structural analysis, and in computer graphics for path calculations.

Math FAQ Answers Dropdown Arrow
Professor Greenline from BrightChamps

Glossary for Cubic Equation Formulas

  • Cubic Equation: A polynomial equation of degree three, typically written in the form ax^3 + bx^2 + cx + d = 0.

     
  • Factorization: A method of solving equations by expressing them as a product of their factors.

     
  • Rational Root Theorem: A theorem used to identify potential rational roots of a polynomial equation.

     
  • Cardano’s Method: A technique for solving cubic equations through transformations and algebraic manipulations.

     
  • Synthetic Division: A simplified form of polynomial division, used to divide polynomials and find roots efficiently.
Math Teacher Background Image
Math Teacher Image

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Max, the Girl Character from BrightChamps

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.

INDONESIA - Axa Tower 45th floor, JL prof. Dr Satrio Kav. 18, Kel. Karet Kuningan, Kec. Setiabudi, Kota Adm. Jakarta Selatan, Prov. DKI Jakarta
INDIA - H.No. 8-2-699/1, SyNo. 346, Rd No. 12, Banjara Hills, Hyderabad, Telangana - 500034
SINGAPORE - 60 Paya Lebar Road #05-16, Paya Lebar Square, Singapore (409051)
USA - 251, Little Falls Drive, Wilmington, Delaware 19808
VIETNAM (Office 1) - Hung Vuong Building, 670 Ba Thang Hai, ward 14, district 10, Ho Chi Minh City
VIETNAM (Office 2) - 143 Nguyễn Thị Thập, Khu đô thị Him Lam, Quận 7, Thành phố Hồ Chí Minh 700000, Vietnam
UAE - BrightChamps, 8W building 5th Floor, DAFZ, Dubai, United Arab Emirates
UK - Ground floor, Redwood House, Brotherswood Court, Almondsbury Business Park, Bristol, BS32 4QW, United Kingdom