Cubic Equation

with

Integer Co-efficient

( only for real+complex roots; for real roots this will fail. click here)

           
x3 + x2 + x + = 0    
ax3    + bx2  + cx  + d  = 0    
   t3      + (depressed cubic equation) (p)*t + (q)    = 0    
t=x  + b/3a   p=(3ac-b2) / 3a2 q=(2b3 -9abc+27a2d) / 27a3      
real root x1= discriminant of depressed cubic : dr1=-(4p3 +27q2   ) = a4*dr1=    
  discriminant of  cubic : 18abcd-4b3d+b2c2-4ac3-27a2d2 Δ0=

Δ1=

C  =

   
nature of roots:     Δ3aa=    
complex/real root x2= + i   Δ'0=    
complex/real root x3= + i   Δ'1=    
      Δ3aa /aΔ3aa    ==22*93 D=aΔ3aa=    
Test the Equation by putting the value  of      x=        
a1x3    + b1x2  + c1x  + d1  =      
* + *  + *  + =     
f'(x) = 3a1x2             + 2b1x                           + c1                   =      
* + *  +   +                   =     
f " (x)=6a1x              + 2b1                     =      
* +                       =    
             
             

 

*The cubic equation with real co-efficient will have 3 roots. These roots are either all real or 1 real, 2 complex with one being the complex conjugate of another.

* Let the equation be ax3    + bx2    +  cx + d =0

* to find the roots , let us calculate the following:-

Δ0=  b2  - 3ac ; when a=1, Δ0=  b2  - 3c

Δ'0=  - Δ0/9 =c/3 - b2/9 (when a=1)

Δ1=  2b3  - 9abc + 27a2d ; when a=1, Δ1=  2b3  - 9bc + 27d

Δ'1=  - Δ1/54=(b/3)*(c/2) -(b/3)3 - d/2 (when a=1)

D=Δ'30+Δ'21 ;

if D < 0, all roots are real, distinct

if D = 0, all roots real, at least two are same.

if D > 0, one root real, rest two are complex conjugates of each other.

del3a=√(Δ21 -4Δ30)

C = ∛[(Δ1 + √(Δ21 -4Δ30))/2]

x1=(1/-3a)(b+ C + Δ0 /C)

x(2&3) =b  +  ξ2C   +  Δ0 /ξ2C   where    ξ=(-1+√3i) /2   and      ξ2 =-(1+√3i) / 2;

x2=(b - C/2 - Δ0/2C ) + (C2 -Δ0 )i /(2√3aC) ;

x3=(b - C/2 - Δ0/2C ) - (C2 -Δ0 )i /(2√3aC) ;

* If discriminant of the cubic > 0 , then there are 3 distinct real roots.

   If discriminant of the cubic < 0 , then there is 1 real and 2 complex roots , one being the complex conjugate of the other.

   If discriminant of the cubic = 0 , then all real & multiple roots.

* In depressed cubic equation, ti = xi + b/3a  where i=1,2,3