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:

Depressing the Cubic Equation

\(\begin\large x= y-\frac\end \) in the above cubic equation, then we get,

\(\begin\large a\left ( y-\frac \right )^+b\left ( y-\frac \right )^+c\left ( y-\frac \right )+d=0\end \)

Simplifying further, we obtain the following depressed cubic equation –

\(\begin\large ay^+\left ( c-\frac> \right )y+\left ( d+\frac<2b^>> +\frac\right )=0\end \)

It must have the term in x 3 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:

Solved Examples on Cubic Equation Formula

Question 1: Solve x 3 – 6x 2 + 11x – 6 = 0
Solution: This equation can be factorized to give

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

Question 2: Solve the cubic equation x 3 – 23x 2 + 142x – 120.

Solution: First factorize the polynomial to get;

x 3 – 23x 2 + 142x – 120 = (x – 1) (x 2 – 22x + 120)

But x 2 – 22x + 120 = x 2 – 12x – 10x + 120

= x (x – 12) – 10(x – 12)

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

Equate each factor to zero to get;

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