# Cubic Equation Formula

The cubic equation formula expresses the cubic equation in Mathematics. An equation with degree three is called a cubic equation. The nature of roots of all cubic equations is either one real root and two imaginary roots or three real roots. If the polynomials have degree three, they are known as cubic polynomials.

## What is Cubic Equation Formula?

To plot the curve of a cubic equation, we need cubic equation formula. This formula helps to find the roots of a cubic equation. If the degree of the polynomial is n, then there will be n number of roots. The roots of cubic equation are also called zeros.

The  cubic  equation formula is given by:

$\LARGE ax^{3}+bx^{2}+cx+d=0$

### Depressing the Cubic Equation

Substitute

$$\begin{array}{l}\large x= y-\frac{b}{3a}\end{array}$$

in the above cubic equation, then we get,

$$\begin{array}{l}\large a\left ( y-\frac{b}{3a} \right )^{3}+b\left ( y-\frac{b}{3a} \right )^{2}+c\left ( y-\frac{b}{3a} \right )+d=0\end{array}$$

Simplifying further, we obtain the following depressed cubic equation –

$$\begin{array}{l}\large ay^{3}+\left ( c-\frac{b^{2}}{3a} \right )y+\left ( d+\frac{2b^{3}}{27a^{2}} +\frac{bc}{3a}\right )=0\end{array}$$

It must have the term in x3 or it would not be cubic ( and so a≠0), but any or all of b, c and d can be zero. For instance:

The examples of cubic equations are:

• x– 6x+ 11x – 6 = 0
• 4x+ 57 = 0
• x+ 9x = 0

## Solved Examples on Cubic Equation Formula

Question 1: Solve x3 – 6x2 + 11x – 6 = 0
Solution: This equation can be factorized to give

(x-1)(x-2)(x-3)=0

This equation has three real roots, all different – the solutions are x = 1, x = 2 and x = 3.

Question 2: Solve the cubic equation x3 – 23x2 + 142x – 120.

Solution:  First factorize the polynomial to get;

x3 – 23x2 + 142x – 120 = (x – 1) (x2 – 22x + 120)

But x2 – 22x + 120 = x2 – 12x – 10x + 120

= x (x – 12) – 10(x – 12)

= (x – 12) (x – 10)

Therefore, x3 – 23×2 + 142x – 120 = (x – 1) (x – 10) (x – 12)

Equate each factor to zero to get;

x = 1

x = 10

x = 12

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