November 1, 2010

Aim: Solve cubics of form


Do Now: Solve by "guess and check"
1)


2)


3)


4) Cubics must have at least 1 root, root(s) for this equation are irrational (none of the potential roots work)



Refer to Solving the Depressed Cubic

In contrast to the quadratic formula,





there is also a cubic formula (Cardano's Formula) developed by Cardano and his student Ferrari.





Solving the Depressed Cubic



This formula only works if (f/3)^3 is less than (g/2)^2

Explanation in words : The solution will always take the form above. The constant under the radical is one half the g term in the equation and the other part under the radical is obtained by subtracting (f/3)^3 from (g/2)^2. The second radical is the same as the first but with the opposite sign under the radical.

Euler's Method: (when given 3ab=f and a^3+b+3=g)





We know that : 3ab = 6 and a^3+b^3=40
(b = 6/3a => b = 2/a)



Using the quadratic formula, we end up with...



To solve for b...



Check :



Gary Wu
Period 3