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.
Solving Cubic Equations
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
.
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
. (Note: "
" means any cube root.