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 $$x$$.

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

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

$$\left( x + \frac{b}{3a} \right)^3 = \frac{b^3 - 27 a^2 d}{27a^3}$$ 6.) take the cubic root of both sides of the equation

$$\left( x + \frac{b}{3a} \right) = \frac{\sqrt[3]{b^3 - 27 a^2 d}}{3a}$$ 7.) subtract $$\frac{b}{3a}$$ from both sides of the equation

$$x = \frac{\sqrt[3]{b^3 - 27 a^2 d}}{3a} - \frac{b}{3a}$$ 8.) simplify the equation

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

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

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