Question

Use Cardano’s algebra method to find the roots of 
x^3+6x^2+9x+4=0

Accepted Answer

Cardano's method applies to what is called the depressed cubic, which is of the form:
t^3+pt+q=0 
To get the equation into the depressed cubic, use the transformation x=t-6/3=t-2 to get:
(t-2)^3+6(t-2)^2+9(t-2)+4   expand
=t^3-6t^2+12t-8    collect like terms
+6t^2-24t+24 
+9t-18+4 
=t^3-3t+2 
This means that the cubic equation to solve using Cardano's method is:
t^3-3t+2=0 
By inspection, we can see that this equation can be factored as
(t+2)(t-1)^2=0 
This equation has solutions t=-2 and t=1 which means that x=-4 and x=-1 .
Cardano's method requires us to use the transformation t=u+v , which ultimately leads to a very intimidating expression for one of the roots:
t_1=(-q/2+sqrt{q^2/4+p^3/27})^{1/3}+(-q/2-sqrt{q^2/4+p^3/27})^{1/3} 
however, in this case, we have the very pleasant condition 4q^3+27p^2=0 which means that the roots are:
t_1=t_2=-{3q}/{2p}=-{3(2)}/{2(-3)}=1 
and 
t_3={3q}/p={3(2)}/-3=-2  which is the same as the method of inspection earlier.
The solution to the cubic equation is x=-4 and x=-1 .

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Use Cardano’s algebra method to find the roots of `x^3+6x^2+9x+4=0`

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