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

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 z¹ terms. Then, multiplying by z³ 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.

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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 z³: 6:54
• Recall PQ version of the Quadratic Formula: 7:54
• Apply the PQ quadratic formula to our equation: 9:51
• Solution for z³: 10:58

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