Cubic Equation
A cubic equation is an equation of the form:
.
A cubic equation has 3 roots, either all real OR one real, two complex.
Contents
Solving Cubic Equations
If you're too lazy to follow, look at subsection "TLDR" for each section.
Converting to a Depressed Equation
You start with the equation .
Divide both sides by a: .
Now we change the coefficient of to
(e.g. change it to a depressed cubic). We do this by substituting
or
, giving:
.
is
and
is
, so now we have
.
TLDR?
The equation is where
and
.
Solving the Depressed Equation
Now here comes the smart part. Substitute .
The equation becomes . Simplification:
We want that last term to equal , so we can set
. (We can't use
, because then
, which is not necessarily true.) Solving this equation gives us
. If
, then
. We now have a system of equations:
.
We can solve this via the quadratic formula. After and
are obtained, we have
and
.
TLDR?
where u and v are roots of the system
.
The Cubic Formula
The cubic formula can be obtained by using the above method. These are the steps:
The depressed cubic is of the form .
and
are the roots of the system of equations
. We can solve this by substitution:
(We are still using p and q because they might get a little messy if we use p and q in terms of a, b, c, and d.)
(comes from
)
(See? I told you it would be messy.) I'm not going to simplify all that squaring and cubing right now: maybe soon! Also, if you select for the first equation and
for the second (or vice versa--they lead to the same number), you will always get a real number.
One last piece of advice: Don't try to memorize this. Memorize the process (shortcut: just look at TLDR for each section). Here is another way to do it.
If you're just asking for the formula for a monic cubic...
Here is the formula for :
If you're asking for the formula for a depressed monic cubic...
Here is the formula for :