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a-b-c-R-ax-4-bx-3-cx-2-bx-a-0-sulation-set-of-the-equation-




Question Number 123451 by Fikret last updated on 25/Nov/20
a,b,c∈R  ax^4 +bx^3 +cx^2 +bx+a=0  sulation set of the equation?
a,b,cRax4+bx3+cx2+bx+a=0sulationsetoftheequation?
Answered by TANMAY PANACEA last updated on 25/Nov/20
ax^2 +(a/x^2 )+bx+(b/x)+c=0  a(x^2 +(1/x^2 ))+b(x+(1/x))+c=0  x+(1/x)=y  a(y^2 −2)+b(y)+c=0  ay^2 +by+c−2a=0  y=((−b±(√(b^2 −4a(c−2a))) )/(2a))=m and  n  m=considering +ve sign before (√ ) sign  n=considering −ve sign before (√ ) sign  x+(1/x)=m  x^2 −mx+1=0  x=((m±(√(m^2 −4)))/2)  x=((n±(√(n^2 −4)))/2)
ax2+ax2+bx+bx+c=0a(x2+1x2)+b(x+1x)+c=0x+1x=ya(y22)+b(y)+c=0ay2+by+c2a=0y=b±b24a(c2a)2a=mandnm=considering+vesignbeforesignn=consideringvesignbeforesignx+1x=mx2mx+1=0x=m±m242x=n±n242
Commented by MJS_new last updated on 25/Nov/20
good!
good!
Commented by TANMAY PANACEA last updated on 25/Nov/20
thank you sir
thankyousir
Answered by Bird last updated on 25/Nov/20
e ⇒x^(4 )  +(b/a)x^3  +(c/a)x^2  +(b/a)x +1=0  (a≠0) equation at form  x^4  +αx^3  +βx^(2 ) +αx +1 =0⇒  x^2  +αx +β +(α/x) +(1/x^2 )=0 ⇒  (x+(1/x))^2 −2+α(x+(1/x))+β=0  we put x+(1/x)=z ⇒  z^2 +αz+β−2=0  Δ=α^2 −4(β−2) =α^2 −4β+8  Δ>0 ⇒z_1 =((−α+(√(α^2 −4β+8)))/2)  z_2 =((−α−(√(α^2 −4β+8)))/2)  after we solve x+(1/x)=z_i   if Δ<0⇒z_1 =((−α+i(√(−Δ)))/2)  z_2 =((−α−i(√(−Δ)))/2)....
ex4+bax3+cax2+bax+1=0(a0)equationatformx4+αx3+βx2+αx+1=0x2+αx+β+αx+1x2=0(x+1x)22+α(x+1x)+β=0weputx+1x=zz2+αz+β2=0Δ=α24(β2)=α24β+8Δ>0z1=α+α24β+82z2=αα24β+82afterwesolvex+1x=ziifΔ<0z1=α+iΔ2z2=αiΔ2.

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