Cubic Equation Formula

Cubic equation is defined by the Cubic Equation Formula. A cubic polynomial, or cubic equation, is a polynomial of degree three. Such equations can have up to three real roots and always have at least one real root. While some of the roots of a cubic equation can be imaginary, there is guaranteed to be at least one real root. Here is an explanation of the Cubic Equation Formula along with a few examples and their solutions.

What is Cubic Equation Formula?

Cubic equation formula can be applied to derive the curve of a cubic equation, making it particularly useful for finding the roots of such equations. A polynomial of degree n will have n zeros or roots. The standard form of a cubic equation is:

ax3+bx2+cx+d=0

In mathematics, a cubic equation is expressed using the Cubic Equation Formula, which defines polynomials of degree three. All cubic equations have roots that can either be one real root and two imaginary roots or three real roots. These three-degree polynomials are known as cubic polynomials.

To solve a cubic equation, it is often helpful to first transform it into a quadratic equation. This quadratic equation can then be factored or solved using the quadratic formula.

While quadratic equations can have up to two real roots, cubic equations can have up to three real roots. Unlike quadratic equations, which may sometimes have no real solutions, cubic equations always have at least one real root. The remaining two roots can be either real or imaginary..

Solving the Cubic Equation

To simplify a cubic equation, we substitute x=y− b/3a into the original cubic equation. This transformation helps us obtain a simpler form of the equation, called a depressed cubic equation.

Starting with the original cubic equation and substituting  x=y− b/3a , we get:

a(y− b/3a)3+b(y− b/3a )2+c(y− b/3a )+d=0

By simplifying this expression, we obtain the following depressed cubic equation:

y3 +py+q=0

Here, the term y3 remains, ensuring it is still a cubic equation (so a≠0). However, any or all of the coefficients p and q can be zero.

Cubic Equation’s General Form

Cubic equation has the general form ax3+bx2+cx+d=0 where a,b,c, and d are constants with 𝑎≠0, and x is the variable. This equation will have three solutions, which may be equal or distinct.

We can solve a cubic equation using two different methods:

• Trial and error combined with synthetic division
• Factorization

Examples Using Cubic Equation Formula

Solved Examples on CubicEquation Formula

Example 1: Solve x3−6x2+11x−6=0.

Solution:

This equation can be factorized as follows:

(x−1)(x−2)(x−3)=0

This factorization indicates that the equation has three distinct real roots:

x=1, x=2, and x=3.

Example 2: Solve the cubic equation x3−23x2+142x−120.

Solution:

First, factorize the polynomial:

X3−23x2+142x−120 = (x−1)(x2−22x+120)

Next, factorize the quadratic term: x2 −22x+120 = x2 −12x−10x+120

=x(x−12)−10(x−12)

=(x−12)(x−10)

Therefore:

X3−23x2+142x−120 = (x−1)(x−10)(x−12)

Equate each factor to zero to find the roots:

x−1=0⇒x=1

x−10=0⇒x=10

x−12=0⇒x=12

Thus, the roots of the equation are x=1, x=10, and x=12.

Example 3: Using the cubic equation formula, solve the cubic equation x3 – 2x2 – x + 2.

Solution:

x3 – 2x2 – x + 2.

= x2(x – 2) – (x – 2)

= (x2 – 1) (x – 2)

= (x + 1) (x – 1) (x – 2)

We can conclude that,

x = -1, x = 1 and x = 2.

1. What is a Cubic Equation?

A Cubic Equation formula of degree three is called a cubic equation. A cubic equation has the conventional form ax3+bx2+cx+d=0.

2. How can students determine the cubic equation's roots?

The methods for locating the roots of the cubic equation are as follows:

1. Utilizing Factor Lists to Find Integer Solutions
2. Graphical Methodology

3. How many roots may a cubic equation have?

A cubic equation has three roots. The roots of a cubic equation could be in any of the following situations:

1. All three roots could be unique and real.
2. Two of the three roots might be equal and all three could be true.
3. All three roots may exist and be equivalent.
4. The other two roots may not be real, but one of them may be.

4. What role does the discriminant play in cubic equations?

The discriminant (Δ) in cubic equations helps determine the nature of their roots. If Δ>0, the equation has three real roots. If Δ=0, it possesses a multiple root and a single real root. If Δ<0, the equation has one real root and two complex conjugate roots.