How are cubic equations converted into this form?

[autodidude]

y = a(x-b)^3 + c

I'm not sure what it's called, my book doesn't mention on how it's derived.

 

[Char. Limit]

Well, not all cubic equations can be converted into this form. For example, x³+4x²+3x+1 can only be converted into (x+1)³+x², and unless you let c be non-constant, it can't be put into the form you're looking for.

 

[HallsofIvy]

I guess it would be called "completing the cube" like "completing the square".

But to be able to "complete the square" you must be given coefficients of $x^2$ and $x$ while to "complete the cube" you would have to be given coefficients of $x^3$, $x^2$, and x. And, as Char.Limit said, unlike completing the square, if those coefficients are not related correctly, it will be impossible. Which is probably why there is not standard name for the operation!