Solving Cubic Equations
edit
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.