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x-4-17-18-x-2-40-3-x-1625-144-0-Find-roots-Two-are-real-and-two-are-complex-




Question Number 201941 by ajfour last updated on 15/Dec/23
x^4 −((17)/(18))x^2 +((40)/3)x+((1625)/(144))=0  Find roots. (Two are real and two   are complex)★
x41718x2+403x+1625144=0Findroots.(Twoarerealandtwoarecomplex)
Commented by Frix last updated on 17/Dec/23
I use the common method.  x^4 +px^2 +qx+r=0  matching with  (x^2 −ux−v)(x^2 +ux−w)=0  leads to  u=(√(s−((2p)/3)))  where s is a solution of  s^3 −(((p^2 +12r)s)/3)−((2p^3 −72pr+27q^2 )/(27))=0  v=−(p/2)−(q/(2u))+(u^2 /2)  w=−(p/2)+(q/(2u))+(u^2 /2)  [if there′s no useable s better approximate]    In the given case  p=−((17)/(18))     q=((40)/3)     r=((1625)/(144))  ⇒  s=((226)/(27))  u=3     v=−((25)/4)     w=−((65)/(36))  ⇒  x=−((13)/6)∨x=−(5/6)∨=(3/2)±2i    How does your new method work?
Iusethecommonmethod.x4+px2+qx+r=0matchingwith(x2uxv)(x2+uxw)=0leadstou=s2p3wheresisasolutionofs3(p2+12r)s32p372pr+27q227=0v=p2q2u+u22w=p2+q2u+u22[iftheresnouseablesbetterapproximate]Inthegivencasep=1718q=403r=1625144s=22627u=3v=254w=6536x=136x=56=32±2iHowdoesyournewmethodwork?
Answered by ajfour last updated on 17/Dec/23
x^4 +bx^2 +cx+d=0  (2x^2 +px+h)(2x^2 −px+k)=3x^2 (x^2 +m)  ⇒  p^2 {(p^2 +b)^2 −4d}=c^2     say   p^2 =p_i ^2     (if three roots  i=0,1,2)       or   p^2 =p_0 ^2    (if one root real)  p^2 +m=−b  x=((±p_0 )/2)±(√(−((c/(±2p_0 ))+(p^2 /4)+(b/2))))  All four roots obtained from any  p=±p_i   .  This example    p_0 =±3  (x∓(3/2))^2 =−[(1/(±6))(((40)/3))+(9/4)−((17)/(36))]   (x∓(3/2))^2 =(4/9), −4  This method really never fails.  because at least one p_i ^2 >0 always. As  p^2 {(p^2 +b)^2 −4d}−c^2 =0  Its been 2 nights, i developed this.
x4+bx2+cx+d=0(2x2+px+h)(2x2px+k)=3x2(x2+m)p2{(p2+b)24d}=c2sayp2=pi2(ifthreerootsi=0,1,2)orp2=p02(ifonerootreal)p2+m=bx=±p02±(c±2p0+p24+b2)Allfourrootsobtainedfromanyp=±pi.Thisexamplep0=±3(x32)2=[1±6(403)+941736](x32)2=49,4Thismethodreallyneverfails.becauseatleastonepi2>0always.Asp2{(p2+b)24d}c2=0Itsbeen2nights,idevelopedthis.
Commented by Frix last updated on 17/Dec/23
From my version: s^3 −(p^2 +12r)s/3−(2p^3 −72pr+27q^2)/27=0 u=√(s−2p/3) u^6 +2pu^4 +(p^2 −4r)u^2 −q^2 =0 From your version: p^2 ((p^2 +b)^2 −4d)−c^2 =0 p^6 +2bp^4 +(b^2 −4d)p^2 −c^2 =0 ...so it′s basically the same... ������
Answered by ajfour last updated on 17/Dec/23
Commented by ajfour last updated on 17/Dec/23
Shouldn't i be awarded for this?

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