The Special Cubic Formula

Part I: The Special Cubic Formula

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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

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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

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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

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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