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

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lfryerda eNotes educator| Certified Educator

Cardano's method applies to what is called the depressed cubic, which is of the form:


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




This means that the cubic equation to solve using Cardano's method is:


By inspection, we can see that this equation can be factored as


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:


however, in this case, we have the very pleasant condition `4q^3+27p^2=0` which means that the roots are:



`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` .