The Special Cubic Formula

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Part I: The Special Cubic Formula[edit]

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 .

Part II: Derivation of the Special Cubic Formula[edit]

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

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

Part III: Limitations of the Formula[edit]

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.

Part IV: Examples[edit]

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