Question Number 114720 by Algoritm last updated on 20/Sep/20

Answered by 1549442205PVT last updated on 20/Sep/20

x^4 −2x^3 +4x−2⇔(x^2 −x+1)^2 −3x^2 +6x−3=0  ⇔(x^2 −x+1)^2 −3(x−1)^2 =0  ⇔[x^2 +((√3)−1)x+1−(√3)][x^2 −(1+(√3))x+1+(√3)]=0  i)⇔x^2 +((√3)−1)x+1−(√3)=0  Δ=((√3)−1)^2 +4((√3)−1)=2(√3)  x=((1−(√3)±(√(2(√3))))/2)  ii)x^2 −(1+(√3))x+1+(√3)=0  Δ=(1+(√3))^2 −4(1+(√3))=−2(√3)<0  ⇒has no roots  Thus,the given equation has two  roots x∈{((1−(√3)+(√(2(√3))))/2),((1−(√3)−(√(2(√3))))/2)}

Answered by MJS_new last updated on 20/Sep/20

x^4 −2x^3 +4x−2=0  (x^2 −(1−(√3))x+1−(√3))(x^2 −(1+(√3))x+1+(√3))=0  x_(1, 2) =((1−(√3)±(√(2(√3))))/2)  x_(3, 4) =((1+(√3))/2)±((√(2(√3)))/2)i