Cubic Formula Proof

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Published: 24 Jan 2025 › Updated: 24 Jan 2025 Cubic Formula Proof

Cubic Formula Proof

In this video I go over a complete derivation of the cubic formula, which is a solution to the cubic equation. This proof utilizes the PQ substitution method, which I first demonstrate by solving for the quadratic formula. Using this substitution method, we can write the cubic equation in the form that we can then apply Vieta's substitution, which converts it into a quadratic equation. Now we can start going backwards and putting our solutions back into the previous substitutions. This involves a cube root, which gives 3 solutions and which we determine via the cube root of unity. Two of the solutions involve complex numbers, but fortunately the resulting 3 solutions in the cubic formula simply interchange the positive and negative signs of the complex numbers. While there is a ton of algebra, many terms repeat themselves so I just copied and pasted to save time. If you like math proofs, you'll love this one!

Note that I followed derivation by Hive account@blackpenredpen :

Time stamps

Intro: 0:00
Recap on Quadratic equation and formula: 0:40
Cubic Formula: 3:58
Derivation of Cubic Formula: 5:46
Step 0: Solve Quadratic Formula using PQ Substitution: 6:00
Step 1: Get rid of square power term using PQ Substitution: 12:44
- Recap on Pascal's Triangle: 18:39
- Applying the substitution: 20:22
Step 2: Apply Vieta's Substitution to obtain a quadratic equation: 28:04
Step 3: Find First Solution of y: 39:17
Step 4.1: Cube Root of Unity: 48:24
- Complex factors are rotations in the complex plane: 1:00:10
Step 4.2: Find the other solutions for y: 1:05:26
Step 5: Put Everything Together to solve for x: 1:13:31
- Double check and summary: 1:25:37
Outro: 1:27:27

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Recap on the Quadratic Formula

Recall my earlier video on the Quadratic Formula:

In that video I derived it by Completing the Square.

Cubic Formula

The cubic formula is the solution to the cubic equation:

where: a ≠ 0

Derivation of Cubic Formula

I will be following the steps done by blackpenredpen's derivation.

Step 0: Solve Quadratic Formula using the PQ Substitution Method

Step 1: Get rid of x² term using PQ Substitution

Just as in the PQ substitution method for the quadratic equation, we would like to get rid of the x² term from the cubic function:

ax³ + bx² + cx + d = 0

It's possible to solve the equation y3 + py + q = 0 by applying Vieta's substitution to obtain a quadratic formula:

Let's now apply the PQ substitution method to the cubic function.

Step 2: Solve y³ + py + q = 0 using Vieta's Substitution

Now we can apply Vieta's substitution to transform our PQ cubic equation into a quadratic equation.

Step 3: Find First Solution

Note that if we plugged in the negative version of the Vieta quadratic formula, we would just get the positive conjugate, and hence the same solution.

Step 4.1: Cube Root of Unity

The cube root gives 3 solutions just as the square root gives 2 solutions.

We can find these solutions by factoring the square and cube root of unity via difference of squares and difference of cubes formula, respectively.

Note also that these complex factors are just rotations in the complex plane:

Step 4.2: Find the other Solutions for y

Step 5: Put Everything Together

Double check:

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