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

In this video I go over the first step in deriving the Cubic Formula, which is to eliminate the x² 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 x² terms get eliminated. Then rearranging and defining p and q as the terms for the coefficients of the x¹ and x⁰, we obtain the PQ version of the Cubic Equation.

Time stamps

• Step 1: Get rid of x² 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 y² terms only: 3:20
• Equate y² 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

Notes and playlists

• Summary: @mes/wwtmttrs
• Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0EF05ExjzLbB64NgUjoV1hl
• Notes: @mes/dzekfnxh .

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