Cubic Formula Proof Step 1: Removing x^2 term via PQ Substitution

mes (67)in DTube • 11 days ago

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

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