# The Special Cubic Formula

## Part I: The Special Cubic Formula

This article discusses a way to solve special cubic equations in the form of

${\displaystyle ax^{3}+bx^{2}+cx+d=0,\quad {\text{where }}c={\frac {b^{2}}{3a}}}$

If the cubic equation satisfies that condition, then you can use the special cubic formula to find the value of ${\displaystyle x}$.

${\displaystyle x={\frac {-b+{\sqrt[{3}]{b^{3}-27a^{2}d}}}{3a}}}$

## Part II: Derivation of the Special Cubic Formula

start with ${\displaystyle ax^{3}+bx^{2}+cx+d=0}$

1.) subtract ${\displaystyle d}$ from both sides of the equation and divide both sides by ${\displaystyle a}$

${\displaystyle x^{3}+{\frac {b}{a}}x^{2}+{\frac {c}{a}}x=-{\frac {d}{a}}}$

2.) find the value of ${\displaystyle k}$ so that

${\displaystyle x^{3}+3kx^{2}+3k^{2}x+k^{3}=(x+k)^{3}}$

There’s a problem with this that puts a limitation on the values of ${\displaystyle b}$ and ${\displaystyle c}$.

${\displaystyle {\frac {b}{3a}}}$ must equal ${\displaystyle {\sqrt {\frac {c}{3a}}}}$ and thus ${\displaystyle \quad c={\frac {b^{2}}{3a}}}$ for the formula to work.

If this condition is true, then the value of ${\displaystyle k}$ is ${\displaystyle {\frac {b}{3a}}}$

3.) add ${\displaystyle \left({\frac {b}{3a}}\right)^{3}}$ (which is ${\displaystyle k^{3}}$) to both sides of the equation

${\displaystyle x^{3}+{\frac {b}{a}}x^{2}+{\frac {c}{a}}x+{\frac {b^{3}}{27a^{3}}}=-{\frac {d}{a}}+{\frac {b^{3}}{27a^{3}}}}$

4.) factor the left side of the equation

${\displaystyle \left(x+{\frac {b}{3a}}\right)^{3}=-{\frac {d}{a}}+{\frac {b^{3}}{27a^{3}}}}$

5.) rearrange the right side of the equation

${\displaystyle \left(x+{\frac {b}{3a}}\right)^{3}={\frac {b^{3}-27a^{2}d}{27a^{3}}}}$

6.) take the cubic root of both sides of the equation

${\displaystyle \left(x+{\frac {b}{3a}}\right)={\frac {\sqrt[{3}]{b^{3}-27a^{2}d}}{3a}}}$

7.) subtract ${\displaystyle {\frac {b}{3a}}}$ from both sides of the equation

${\displaystyle x={\frac {\sqrt[{3}]{b^{3}-27a^{2}d}}{3a}}-{\frac {b}{3a}}}$

8.) simplify the equation

${\displaystyle x={\frac {-b+{\sqrt[{3}]{b^{3}-27a^{2}d}}}{3a}}}$

## Part III: Limitations of the Formula

As stated above, this formula can only be used in special cases where ${\displaystyle c}$ and ${\displaystyle b}$ are dependent on each other. The equations that display this are:

${\displaystyle c={\frac {b^{2}}{3a}}}$  or equivalently  ${\displaystyle b=\pm {\sqrt {3ac}}}$

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

Example 1:  ${\displaystyle 3x^{3}+6x^{2}+4x+9=0}$

Step 1: Check if the equation obeys the limitations

${\displaystyle c=4\quad =\quad {\frac {b^{2}}{3a}}={\frac {6^{2}}{3\cdot 3}}=4}$

Step 2: Since the equation obeys the criteria of a special cubic equation, the special cubic formula may be applied

${\displaystyle x={\frac {-6+{\sqrt[{3}]{6^{3}-27\cdot 3^{2}\cdot 9}}}{3\cdot 3}}={\frac {-6-{\sqrt[{3}]{1971}}}{9}}\approx -2.05978}$

${\displaystyle 3(-2.05978)^{3}+6(-2.05978)^{2}+4(-2.05978)+9\approx 0}$

Example 2:  ${\displaystyle 3x^{3}+21x^{2}+2x+3=0}$

Step 1: Check if the equation obeys the limitations

${\displaystyle c=2\quad \neq \quad {\frac {b^{2}}{3a}}={\frac {21^{2}}{3\cdot 3}}=49}$

This equation doesn’t obey the limitations, so it is not a special cubic equation.

Example 3:  ${\displaystyle 3x^{3}-6x^{2}+4x-5=0}$

Step 1: Check if the equation obeys the limitations

${\displaystyle c=4\quad =\quad {\frac {b^{2}}{3a}}={\frac {(-6)^{2}}{3\cdot 3}}=4}$

Step 2: Since the equation obeys the criteria of a special cubic equation, the special cubic formula may be applied

${\displaystyle x={\frac {6+{\sqrt[{3}]{-6^{3}-27\cdot 3^{2}\cdot -5}}}{3\cdot 3}}={\frac {6+{\sqrt[{3}]{999}}}{9}}\approx 1.7774}$

${\displaystyle 3(1.7774)^{3}-6(1.7774)^{2}+4(1.7774)-5\approx 0}$