In this video I go over the third step of the Cubic Formula proof which involves plugging our z value back into Vieta's substitution to obtain our first y solution. Recall that in Step 1 we removed the x-squared term from the cubic equation via PQ substitution, and in Step 2 we used Vieta's substitution to obtain a quadratic formula. For Step 3, and since the previous quadratic formula obtained two solutions, we start by plugging in the positive version into Vieta's substitution. After some algebra via conjugate multiplication, we obtain the first solution for y (before we have to plug it into our PQ substitution to obtain x), which has both the positive and negative versions of the quadratic formula solution. This also means that if plugged in the negative version into Vieta's substitution, we would still obtain the positive, thus it doesn't matter which version we plug in!
Notes: Step 3: Find First Solution
Written notes — mirrored from Cubic Formula Proof, which walks through the complete derivation.
Note that if we plugged in the negative version of the Vieta quadratic formula, we would just get the positive conjugate, and hence the same solution.
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- Full written notes: Cubic Formula Proof
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