# What is auxiliary equation?

1. Dec 16, 2011

### athrun200

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

2. Dec 16, 2011

### 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.

3. Dec 16, 2011

### athrun200

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

4. Dec 16, 2011

### I like Serena

Hmm, not sure.
*looking up calculus*
No, not calculus.

*looking up algebra*
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!

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