Cubic Equation (when D < 0, all roots are real)

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

       
x3 + (b)x2 + (c)x + (d)=0  
D'=Δ'30  + Δ'21 Δ'0=(3c-b2) / 9 Δ'1=(9bc-27d-2b3)/54    
D': Δ'0= Δ'1=    
comment:

   
D=(-4Δ30  + Δ21) Δ0=-(3c-b2) Δ1=-(9bc-27d-2b3) D/D'==93*22 =2916  
   
θ=cos-1  [Δ1/√(-Δ30)] θ1=θ +2π θ2=θ +4π del03a=√(-Δ0)  
θ θ1 θ2 del03a  
x1a: 2√(-Δ0)cos(θ /3) -b/3 x1b: 2√(-Δ0)cos(θ1 /3) -b/3 x1c: 2√(-Δ0)cos(θ2 /3) -b/3    
x1a x1b x1c