The Special Cubic Formula

Part I: The Special Cubic Formula

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

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

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 $a{x}^3+b{x}^2+cx+d=0$

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

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

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 $c=\frac{b^2}{3a}$ 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

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

5.) rearrange the right side of the equation

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

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

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

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:

$c=\frac{b^2}{3a}$ Template:Pador equivalentlyTemplate:Pad $b=\pm \sqrt{3ac}$

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: Template:Pad$3x^3+6x^2+4x+9=0$

Step 1: Check if the equation obeys the limitations

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

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: Template:Pad$3x^3+21x^2+2x+3=0$

Step 1: Check if the equation obeys the limitations

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

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

Example 3: Template:Pad$3x^3-6x^2+4x-5=0$

Step 1: Check if the equation obeys the limitations

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

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

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