Cubic Formula Proof Step 2: Applying Vieta's Substitution to Obtain a Quadratic Equation

mes (67)in DTube • 2 days ago

In this video I go over the second step in deriving the Cubic Formula, and this involves converting our earlier PQ version of the cubic formula into a quadratic formula using Vieta's substitution. In step 1 we converted x into y by the equation x = y + k. In step 2 we convert y into z by the equation y = z + k/z. Then we do the exact same procedure as before to select a k value that cancels out the z^1 terms. Then, multiplying by z^3 we obtain a quadratic equation, which we can solve using the PQ version of the quadratic formula. In later steps, we will convert z back into y and then finally back into x.

Timestamps:

Step 2: Solve PQ Cubic Equation using Vieta's substitution to obtain quadratic formula: 0:00
Let y = z + k/z: 0:57
Select k to cancel out z terms: 3:07
k = -P/3: 4:13
Convert resulting equation into a quadratic equation by multiplying by z^3: 6:54
Recall PQ version of the Quadratic Formula: 7:54
Apply the PQ quadratic formula to our equation: 9:51
Solution for z^3: 10:58

Notes and playlists: