x^4 +ax^2 +bx+c=0 let x=(t/s) ⇒ t^4 +as^2 t^2 +bs^3 t+cs^4 =0 let t=p+h ⇒ p^4 +4hp^3 +6h^2 p^2 +4h^3 p+h^4 +as^2 (p^2 +2hp+h^2 ) +bs^3 (p+h)+cs^4 =0 ⇒ p^4 +4hp^3 +(6h^2 +as^2 )p^2 +(4h^3 +2ahs^2 +bs^3 )p +(h^4 +as^2 h^2 +bs^3 h+cs^4 )=0 let ((√2)p^2 +Ap+B)^2 =(p^2 +mp+k)^2 ⇒ 2(√2)A−2m=4h 2(√2)B+A^2 −m^2 −2k = 6h^2 +as^2 2AB−2mk=4h^3 +2ah+bs^3 B^2 −k^2 =h^4 +as^2 h^2 +bs^3 h+cs^4 let m=0 ⇒ A=(√2)h 2(√2)B=4h^2 +as^2 +2k 4h^3 +ahs^2 +2hk=4h^3 +2ah+bs^3 ⇒ (as^2 +2k−2a)h=bs^3 (4h^3 +as^2 +2k)^2 −8k^2 =8(h^4 +as^2 h^2 +bs^3 h+cs^4 ) let k=a ⇒ ah=bs ⇒ (((4b^2 h)/a^2 )+a+((2a^3 h^2 )/b^2 ))^2 −((8b^4 h^4 )/a^2 ) =8((b^4 /a^4 )+((2b^2 )/a)+c) ⇒ ((16b^4 h^2 )/a^4 )+((4a^6 h^4 )/b^4 ) +((8b^2 h)/a)+32ah^3 +((4a^4 h^2 )/b^2 ) −((8b^4 h^4 )/a^2 )=λ ⇒ (((4a^6 )/b^4 )−((8b^4 )/a^2 ))h^4 +32ah^3 +(((16b^4 )/a^4 )+((4a^4 )/b^2 ))h^2 +((8b^2 h)/a)−λ=0 ...


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Question Number 143822 by ajfour last updated on 18/Jun/21

$x^{4} + {a x}^{2} + b x + c = 0$ $l e t x = \frac{t}{s}$ $\Rightarrow t^{4} + {a s}^{2} t^{2} + {b s}^{3} t + {c s}^{4} = 0$ $l e t t = p + h \Rightarrow$ $p^{4} + 4 {h p}^{3} + 6 h^{2} p^{2} + 4 h^{3} p + h^{4}$ $+ {a s}^{2} (p^{2} + 2 h p + h^{2})$ $+ {b s}^{3} (p + h) + {c s}^{4} = 0$ $\Rightarrow$ $p^{4} + 4 {h p}^{3} + (6 h^{2} + {a s}^{2}) p^{2}$ $+ (4 h^{3} + 2 {a h s}^{2} + {b s}^{3}) p$ $+ (h^{4} + {a s}^{2} h^{2} + {b s}^{3} h + {c s}^{4}) = 0$ $l e t {(\sqrt{2} p^{2} + A p + B)}^{2} = {(p^{2} + m p + k)}^{2}$ $\Rightarrow 2 \sqrt{2} A - 2 m = 4 h$ $2 \sqrt{2} B + A^{2} - m^{2} - 2 k$ $= 6 h^{2} + {a s}^{2}$ $2 A B - 2 m k = 4 h^{3} + 2 a h + {b s}^{3}$ $B^{2} - k^{2} = h^{4} + {a s}^{2} h^{2} + {b s}^{3} h + {c s}^{4}$ $l e t m = 0 \Rightarrow$ $A = \sqrt{2} h$ $2 \sqrt{2} B = 4 h^{2} + {a s}^{2} + 2 k$ $4 h^{3} + {a h s}^{2} + 2 h k = 4 h^{3} + 2 a h + {b s}^{3}$ $\Rightarrow ({a s}^{2} + 2 k - 2 a) h = {b s}^{3}$ ${(4 h^{3} + {a s}^{2} + 2 k)}^{2}$ $- 8 k^{2} = 8 (h^{4} + {a s}^{2} h^{2} + {b s}^{3} h + {c s}^{4})$ $l e t k = a$ $\Rightarrow a h = b s$ $\Rightarrow {(\frac{4 b^{2} h}{a^{2}} + a + \frac{2 a^{3} h^{2}}{b^{2}})}^{2} - \frac{8 b^{4} h^{4}}{a^{2}}$ $= 8 (\frac{b^{4}}{a^{4}} + \frac{2 b^{2}}{a} + c)$ $\Rightarrow \frac{16 b^{4} h^{2}}{a^{4}} + \frac{4 a^{6} h^{4}}{b^{4}}$ $+ \frac{8 b^{2} h}{a} + 32 {a h}^{3} + \frac{4 a^{4} h^{2}}{b^{2}}$ $- \frac{8 b^{4} h^{4}}{a^{2}} = λ$ $\Rightarrow (\frac{4 a^{6}}{b^{4}} - \frac{8 b^{4}}{a^{2}}) h^{4} + 32 {a h}^{3}$ $+ (\frac{16 b^{4}}{a^{4}} + \frac{4 a^{4}}{b^{2}}) h^{2} + \frac{8 b^{2} h}{a} - λ = 0$ $. . .$
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