## Cubic equation solution

on Sun Aug 29, 2010 5:31 pm

Scipione del Ferro (6 February 1465 – 5 November 1526) was an Italian mathematician who first discovered a method to solve the depressed cubic equation.

Mathematicians from del Ferro's time knew that the general cubic equation could be simplified to one of two cases called the depressed cubic equation, for positive numbers p, q and x.

x^3 + px = q

x^3 = q - px

If x = u + v , show how x^3 + px = q has a solution given by

x = (q/2 + ((p/3)^3 + (q/2)^2)^1/2)^1/3 - (-q/2 + ((p/3)^3 + (q/2)^2)^1/2)^1/3

Mathematicians from del Ferro's time knew that the general cubic equation could be simplified to one of two cases called the depressed cubic equation, for positive numbers p, q and x.

x^3 + px = q

x^3 = q - px

If x = u + v , show how x^3 + px = q has a solution given by

x = (q/2 + ((p/3)^3 + (q/2)^2)^1/2)^1/3 - (-q/2 + ((p/3)^3 + (q/2)^2)^1/2)^1/3

_________________

## Re: Cubic equation solution

on Sun Aug 29, 2010 5:35 pm

My math teacher also gave me this damn question. I'm sucked after substituting x=u+v into the original equation.

I got (u^3 + v^3)+(u+v)(3uv+p)=q. But then what?!?

I got (u^3 + v^3)+(u+v)(3uv+p)=q. But then what?!?

## Re: Cubic equation solution

on Sun Aug 29, 2010 5:36 pm

Jump off the window... that's what I would do.marshmallow wrote:My math teacher also gave me this damn question. I'm sucked after substituting x=u+v into the original equation.

I got (u^3 + v^3)+(u+v)(3uv+p)=q. But then what?!?

## Re: Cubic equation solution

on Sun Aug 29, 2010 5:39 pm

You have to prove that the equation is satisfied if (u^3 + v^3)=q and uv=-p/3.marshmallow wrote:My math teacher also gave me this damn question. I'm sucked after substituting x=u+v into the original equation.

I got (u^3 + v^3)+(u+v)(3uv+p)=q. But then what?!?

## Re: Cubic equation solution

on Sun Aug 29, 2010 5:40 pm

And how am I supposed to do that?einsteinium wrote:You have to prove that the equation is satisfied if (u^3 + v^3)=q and uv=-p/3.

## Re: Cubic equation solution

on Sun Aug 29, 2010 5:44 pm

Dud! The level of this question is really too high for me! Even for my dad.

But I figured it out after copying on π3.14159265. Ha ha!

I wonder if he would look life if he sees this.

(But I will not post his answer on Internet without his permission. )

But I figured it out after copying on π3.14159265. Ha ha!

I wonder if he would look life if he sees this.

(But I will not post his answer on Internet without his permission. )

## Re: Cubic equation solution

on Sun Aug 29, 2010 5:49 pm

Ah, come on bro! Who cares? I waited so long for someone to give me the accurate answer.Lollipop wrote:Dud! The level of this question is really too high for me! Even for my dad.

But I figured it out after copying on π3.14159265. Ha ha!

I wonder if he would look life if he sees this.

(But I will not post his answer on Internet without his permission. )

## Re: Cubic equation solution

on Sun Aug 29, 2010 5:51 pm

The answer can be found if you do it in a different way.

Try to substitute x=u+v into x^3 =q-px and you'll be able to give prove for (u^3 + v^3)=q and uv=-p/3.

Try to substitute x=u+v into x^3 =q-px and you'll be able to give prove for (u^3 + v^3)=q and uv=-p/3.

## Re: Cubic equation solution

on Sun Aug 29, 2010 5:55 pm

OMG! You're my savior!π3.14159265 wrote:The answer can be found if you do it in a different way.

Try to substitute x=u+v into x^3 =q-px and you'll be able to give prove for (u^3 + v^3)=q and uv=-p/3.

I found x^3 = (u^3 + v^3)+3uv(u+v) which is the same thing than x^3 =q-px.

Since x=u+v, I concluded that (u^3 + v^3)=q and 3uv=p.

But then? What? TT

## Re: Cubic equation solution

on Sun Aug 29, 2010 5:56 pm

Just give us the complete answer, bro!π3.14159265 wrote:The answer can be found if you do it in a different way.

Try to substitute x=u+v into x^3 =q-px and you'll be able to give prove for (u^3 + v^3)=q and uv=-p/3.

## Re: Cubic equation solution

on Sun Aug 29, 2010 6:06 pm

If you pay me.InfluenzaKiller wrote:Just give us the complete answer, bro!

In the quadratic equation, where ay^2 + by + c = 0 and where y = y1 and y2,

y1+y2=-b/a ----> (u^3 + v^3)=q

y1y2=c/a ----> uv=-p/3

So we can say that u^3 = y1 and v^3 = y2

And that means -b/a=q and c/a=(-p/3)^3

Now substitute them in this equation :

x = y1^1/3 + y2^1/3 = (-b+(b^2 -4ac)^1/2 ÷ 2a)^1/3 + (-b-(b^2 -4ac)^1/2 ÷ 2a)^1/3

And you'll find the solution to the depressed cubic.

## Re: Cubic equation solution

on Sun Aug 29, 2010 6:08 pm

Cool! Thanks! I'll pay you a bus ticket tomorrow.π3.14159265 wrote:

If you pay me.

In the quadratic equation, where ay^2 + by + c = 0 and where y = y1 and y2,

y1+y2=-b/a ----> (u^3 + v^3)=q

y1y2=c/a ----> uv=-p/3

So we can say that u^3 = y1 and v^3 = y2

And that means -b/a=q and c/a=(-p/3)^3

Now substitute them in this equation :

x = y1^1/3 + y2^1/3 = (-b+(b^2 -4ac)^1/2 ÷ 2a)^1/3 + (-b-(b^2 -4ac)^1/2 ÷ 2a)^1/3

And you'll find the solution to the depressed cubic.

## Re: Cubic equation solution

on Sun Aug 29, 2010 6:10 pm

Scipione del Ferro

Hewas born in Bologna, in northern Italy, to Floriano and Filippa Ferro. His father, Floriano, worked in the paper industry, which owed its existence to the invention of the press in the 1450s and which probably allowed Scipione to access various works during early stages of his life. He married and had a daughter, who was named Filippa after his mother.

He likely studied at the University of Bologna, where he was appointed a lecturer in Arithmetic and Geometry in 1496. During his last years, he also undertook commercial work.

Hewas born in Bologna, in northern Italy, to Floriano and Filippa Ferro. His father, Floriano, worked in the paper industry, which owed its existence to the invention of the press in the 1450s and which probably allowed Scipione to access various works during early stages of his life. He married and had a daughter, who was named Filippa after his mother.

He likely studied at the University of Bologna, where he was appointed a lecturer in Arithmetic and Geometry in 1496. During his last years, he also undertook commercial work.

- Spoiler:
- Diffusion of his work

There are no surviving scripts from del Ferro. This is in large part due to his resistance to communicating his works. Instead of publishing his ideas, he would only show them to a small, select group of friends and students.

It is suspected that this is due to the practice of mathematicians at the time of publicly challenging one another. When a mathematician accepted another's challenge, each mathematician needed to solve the other's problems. The loser in a challenge often lost funding or his university position. Del Ferro was fearful of being challenged and likely kept his greatest work secret so that he could use it to defend himself in the event of a challenge.

Despite this secrecy, he had a notebook where he recorded all his important discoveries. After his death in 1526, this notebook was inherited by his son-in-law Hannival Nave, who was married to del Ferro's daughter, Filippa. Nave was also a mathematician and a former student of del Ferro's, and he replaced del Ferro at the University of Bologna after his death.

In 1543, Gerolamo Cardano and Ludovico Ferrari (one of Cardano's students) travelled to Bologna to meet Nave and learn about his late father-in-law's notebook, where the solution to the depressed cubic equation appeared.

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