Last updated on August 5th, 2025
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.
Cubic equations involve polynomials of degree three. Let’s learn the formulas to solve cubic equations using different methods.
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.
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.
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.
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.
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.
Students often make errors when solving cubic equations. Here are some mistakes and ways to avoid them to master these equations.
Solve the cubic equation x^3 - 6x^2 + 11x - 6 = 0?
The roots are x = 1, x = 2, and x = 3
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.
Solve the cubic equation 2x^3 - 4x^2 - 22x + 24 = 0?
The roots are x = -2, x = 2, and x = 3
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.
Find the roots of the cubic equation x^3 + 3x^2 - 4x - 12 = 0?
The roots are x = -3, x = -2, and x = 2
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.
Solve the cubic equation x^3 - 3x^2 - 4x + 12 = 0?
The roots are x = -2, x = 2, and x = 3
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.
Find the roots of the cubic equation x^3 - x^2 - 9x + 9 = 0?
The roots are x = -3, x = 1, and x = 3
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.
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.
: He loves to play the quiz with kids through algebra to make kids love it.