This article discusses a way to solve special cubic equations in the form of
If the cubic equation satisfies that condition, then you can use the special cubic formula to find the value of
.
start with
1.) subtract
from both sides of the equation and divide both sides by
2.) find the value of
so that
There’s a problem with this that puts a limitation on the values of
and
.
must equal
and thus
for the formula to work.
If this condition is true, then the value of
is ![{\displaystyle {\frac {b}{3a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9be227cfeba5821c181dfa5ff820bc82dfd9aef)
3.) add
(which is
) to both sides of the equation
4.) factor the left side of the equation
5.) rearrange the right side of the equation
6.) take the cubic root of both sides of the equation
7.) subtract
from both sides of the equation
8.) simplify the equation
As stated above, this formula can only be used in special cases where
and
are dependent on each other. The equations that display this are:
or equivalently
If the cubic equation in question does not obey these equations, then a much longer formula must be used to find the solution. These two equations also restrict the cubic formula to cubic equations that only have one solution.
Example 1:
Step 1: Check if the equation obeys the limitations
Step 2: Since the equation obeys the criteria of a special cubic equation, the special cubic formula may be applied
Step 3: Check the answer
Example 2:
Step 1: Check if the equation obeys the limitations
This equation doesn’t obey the limitations, so it is not a special cubic equation.
Example 3:
Step 1: Check if the equation obeys the limitations
Step 2: Since the equation obeys the criteria of a special cubic equation, the special cubic formula may be applied
Step 3: Check the answer