How Does a Cubic Equation Simplify to a Quadratic?

[athrun200]

The auxiliary equation for a cubic was simpler—a quadratic. This “resolvent quadratic” could be solved by the Babylonian method; then the solution of the cubic could be reconstructed by taking a cube root. Why Beauty is Truth P78

Can anyone can give one example how cubic become quadratic?
Or can you recommand me some books? (I am an university student.)

 

[I like Serena]

Hi athrun200!

Here's a method where a cubic equation is reduced to a quadratic equation.


Starting with $x^3+3x+6=0$.

Substitute x=y+z, meaning you have a free choice for either y or z.
So $(y+z)^3+3(y+z)+6=0$

$\Rightarrow (y^3+z^3+3y^2z+3yz^2) + 3(y+z)+6=0$

$\Rightarrow y^3+z^3+3(yz+1)(y+z)+6=0$


Choose z such that $yz+1=0$, or $z=-{1 \over y}$
Then: $y^3 - {1 \over y^3} + 6 = 0$
$\Rightarrow (y^3)^2 + 6(y^3) - 1 = 0$


Solve as a quadratic equation and back substituting z gives:
$$x=y+z=\sqrt[3]{-3 + \sqrt{10}} - {1 \over \sqrt[3]{-3 + \sqrt{10}}}$$
or
$$x=y+z=-\sqrt[3]{3 + \sqrt{10}} + {1 \over \sqrt[3]{3 + \sqrt{10}}}$$

Note that both solutions are the same root.

 

[athrun200]

Thx very much!
which topic does it belong to? Calculus? or Algebra?
Are there any textbook about this?

 

[I like Serena]

Hmm, not sure.
*looking up calculus*

Calculus (Latin, calculus, a small stone used for counting) is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series.

No, not calculus.


*looking up algebra*

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures.

Yes, I think it's algebra!


TBH, I learned it from an old book of my father, that he had while studying.
I don't recall what it was called, but it was a thick volume with a purple cover and many yellowed pages... I loved it!