CUBIC EQUATIONS

By: Jenissa Reynoso

Introduction

The following slides explains how to solve a cubic

equation

To start off I’m going to give you a brief cool history of

the cubic equation

Then I’ll explain how to find the depressed cubic

equation which will help get to the solution

History of the cubic equation

The cubic equation goes as far back to 400 B.C, the birth

of this problem came from a geometric question from

Greek mathematicians. The question was the following ,

“Given an angle, is there a way to construct an angle one

third as large?”

When they mentioned construct

they weren’t talking about a

compass and ruler, this type of

construction would require

other tools. In Ancient Greece

they had many constructions

involving conic sections like

parabolas and hyperbolas.

History of the cubic equation

Once trigonometry came along the mathematicians came up with a way of solving the cubic equation. Finding x amounts to solving this equation.

To find one third of the given angle (theta) ᶿ, we can begin by thinking of

(theta) ᶿ as three times the angle we’re looking for, which we’ll call (alpha) α;

that is, α = θ/3 . Now we apply the formula for the cosine of 3α:

Since the angle (theta) ᶿ is known, we also know cos(θ); call it a. To

construct θ/3 , we need to construct its cosine. If we let then using

the formula above with α = θ/3 , we get

History of the cubic equation

After mathematicians knew about this equation

the first to attempt to solve it was Al-

Khayammi, he thought he had it all figured out

but when he tried using actual numbers his

method was no help at all and he admitted to it.

Many mathematicians had a

lot of trouble figuring out how

to solve a cubic equation until

Scipione Del Ferro (1465-

1526) and Niccolo Fontana

(Tartaglia) (1500 – 1557)

from Italy came in the picture.

History of the cubic equation

They both discovered how to solve a cubic equation and kept it a

secret in order to challenge other people. The person Ferro passed

the secret to one of his students Antonio Maria Fiore, who

challenged Tartaglia in a competition. They both had different

ways of solving the cubic equation but were both correct.

History of the cubic equation

However Tartaglia beat Fiore and once the news was out about the

competition Cardano wanted to meet Tartaglia to get the secret

from him which he did after convincing Tartaglia that he would

keep it a secret also. His secrecy didn’t last for long because he

wanted to publish this solution, in order to not break his promise

of secrecy he used Fiore’s way of solving the cubic equation.

History of the cubic equation

Cardano assistant Ferrari figured out how to

solve the equation of degree four (the

quartic) and Cardano published a book called

Arc Magna meaning “The great Art”

revealing the solution to the cubic equation

and quartic equation.

There was one mistake that Cardano had in his

method which was later on resolved by

Bombelli, all he did was use the square root of

negative numbers as possible solutions. The

cubic equation is one of the mathematical

problems that led to the development of

abstract algebra.

How does one find the formula for the solution of the cubic?

To find the solution to the cubic equation we first have to find the

formula in other words we have to find the cubic equation without

a square term sometimes called “depressed cubic”.We start with

the complete cubic equation which is:

How does one find the formula for the solution of the cubic? if the cubic equation doesn’t have one as a leading coefficient just

divide both sides of the equation by that leading coefficient and you’ll end up with one as a leading coefficient. Then we continue with the first substitution:

Let

when we substitute this into the equation we end up with:

How does one find the formula for the solution of the cubic? Now we need to expand these terms:

(Cubic term):

(Quadratic term):

(Linear term):

(Constant):

How does one find the formula for the solution of the cubic? Once we expanded these terms we simplify them:

How does one find the formula for the solution of the cubic? Then we gather like term we end up with:

We substitute again we let and

this is how we get the cubic equation without a square term

sometimes called “depressed cubic” which is:

The solution for the cubic equations Since we have the depressed cubic equation like our little face as

shown, we can now find the solution :

We let

After substituting and simplifying the equation we end up with :

The solution for the cubic equations Then we multiply by which gives us:

this is a quadratic in z^3

We plug into the quadratic formula

Once we find the solution to the quadratic equation we substitute back to get the solution for x.

Extra help!!! If you need any

extra help please feel free to e-mail me at reynosje@kean.edu or ask after class for additional help I’ll be happy to help.

Just remember Math is awesome!!!