Name: Cubic Formula Proof Step 1: Removing x^2 term via PQ Substitution
Uploaded: 2025-02-03T23:46:10+00:00
Duration: 15 min 9 s
Description: In this video I go over the first step in deriving the Cubic Formula, which is to eliminate the x^2 term from the Cubic Equation by using the PQ Substitution method. This allows us to later apply Viet

9 months ago

126

Health & Science Education math science education cubic equation cubic formula complex numbers imaginary numbers cube root of unity mathematics

In this video I go over the first step in deriving the Cubic Formula, which is to eliminate the x^2 term from the Cubic Equation by using the PQ Substitution method. This allows us to later apply Vieta's substitution to obtain a quadratic formula, which we can solve. As in the PQ Quadratic Formula method, I set x = y + k and then expand the powers and select a k value such that the x^2 terms get eliminated. Then rearranging and defining p and q as the terms for the coefficients of the x^1 and x^0, we obtain the PQ version of the Cubic Equation.

Time stamps:

- Step 1: Get rid of x^2 term using PQ Substitution in order to later apply Vieta's substitution to get a quadratic formula: 0:00
- Applying PQ substitution to cubic equation by letting x = y + k: 1:46
- Expanding out the y^2 terms only: 3:20
- Equate y^2 terms to equal zero and solve for k: 5:10
- Recap of Pascal's Triangle: https://en.wikipedia.org/wiki/Pascal%27s_triangle 6:00
- Plugging in our k to cancel out terms and rearrange equation: 7:43
- Divide new Cubic Equation by a: 13:40
- Substitute p and k to obtain the new PQ Cubic Equation: 14:45