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Solving Cubic Equations
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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.