Cubic formula
The cubic formula is a very complicated formula used to solve cubics. It is not used very often, as schools don't teach it and problem writers usually hide a simpler tactic instead.
Cardano's formula
Start with the cubic . The constant
controls how steep it is. The constant
shifts it up and down. The constant
controls the slope of the middle. And the constant
shifts it left to right. For now, let's just reduce it to
, as everyone knows how to do that. Consider the cubic
. The first two terms match, and if we substitute
with
, we get rid of that annoying
term. We'll just add it back at the end. Making the substitution results in:
.