Solve the equation x^4-2x^+4x^2+6x-21=0. Given that the sum of two of its roots is zero

 

          Bs Grewal

         exercise solution

Question:- Solve the equation x^4-2x^3+4x^2+6x-21=0. Given that the sum of two of its roots is zero.

 

Solution:- let 𝛂,𝛃,Ɣ,𝛅 are the roots of the equation such that 

Also, 𝞪+𝛃+Ɣ+𝛅=2    =>Ɣ+𝛅=2

Thus the quadratic factor corresponding to α,β is of the form

X^2-0x+p and that corresponding to   is of the form of x^2-2x+q

Hence,  x^4-2x^3+4x^2+6x-21=(x^2+p)(x^2-2x+q)

 x^4-2x^3+4x^2+6x-21=x^4+(p+q)x^2-2x^3-2xp+pq

 

Equating coefficients of from On both  sides of the above equation, we get

p+q=4     and    -2p=6   => p=-3

and q=7

hence the given equation is equivalent to 

hence the roots x=±√3,1+i√3

solve (x^2 - 4xy -2y^2)dx+(y^2-4xy -2x^2)dy=0