Question Number 201941 by ajfour last updated on 15/Dec/23

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?](https://www.tinkutara.com/question/Q201959.png)
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.](https://www.tinkutara.com/question/Q201962.png)
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...
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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?