Answer to Question #86906 in Algebra for sonali mansingh

Question

Apply Cardano’s method for finding the roots of .0 2x 3x

Answer 1

To apply the Cardano method it is necessary to bring this equation to an canonical form:

y^3+py+q=0

write down our cubic equation in the general form

ax^3+bx^2+cx+d=0

1*x^3+1*x^2+0*x+0=0

We will replace the variable

x=y-(b/3a)

by changing the variable can be reduced to the above canonical form with coefficients

p=c/a-(b^2/3a^2)=-1/3

q= (2b^3/27a^3)-(bc/3a^2)+d/a=2/27

the canonical form will have this form

y^3-1/3y+2/27=0

Determine the value

Q=(p/3)^3+(q/2)^2=-1/729+1/729=0

Q = 0 is a single real real root

y=a+b

where

a=((-q/2)+Q^½)^⅓

b=((-q/2)-Q^½)^⅓

y=(-1/3)+(-1/3)=-2/3

then

x= (-2/3)-(1/3)=-1