The Special Cubic Formula

Part I: The Special Cubic Formula

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

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

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

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

Part II: Derivation of the Special Cubic Formula

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

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

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

2.) find the value of $k$ so that

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

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

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

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

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

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

4.) factor the left side of the equation

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

5.) rearrange the right side of the equation

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

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

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

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

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

8.) simplify the equation

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

Part III: Limitations of the Formula

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

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

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:  $3x^{3}+6x^{2}+4x+9=0$

Step 1: Check if the equation obeys the limitations

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

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

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

Step 3: Check the answer

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

Example 2:  $3x^{3}+21x^{2}+2x+3=0$

Step 1: Check if the equation obeys the limitations

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

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

Example 3:  $3x^{3}-6x^{2}+4x-5=0$

Step 1: Check if the equation obeys the limitations

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

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

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

Step 3: Check the answer

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