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Solving Cubic Equations

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Details showing that the two forms are the same

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The two plans for solving a cubic equation give what looks like two different solutions. In both plans the formulas for P, Q, R, and S are the same.

Starting from

Cardano's method result is:

Vieta's method result is:

These two can be shown to be the same if the second terms of each are identical.

If

Starting with:

Multiply by a unit fraction:

Carry through the sum and difference product in the denominators:

Add like terms in the denominator:

Finally cancel like factors in the numerator and the demoninator:

So the two methods yield the identical results.