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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-

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Question Number 143822 by ajfour last updated on 18/Jun/21
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 ...
$$\:{x}^{\mathrm{4}} +{ax}^{\mathrm{2}} +{bx}+{c}=\mathrm{0} \\ $$$${let}\:\:{x}=\frac{{t}}{{s}} \\ $$$$\Rightarrow\:{t}^{\mathrm{4}} +{as}^{\mathrm{2}} {t}^{\mathrm{2}} +{bs}^{\mathrm{3}} {t}+{cs}^{\mathrm{4}} =\mathrm{0} \\ $$$${let}\:{t}={p}+{h}\:\:\Rightarrow \\ $$$${p}^{\mathrm{4}} +\mathrm{4}{hp}^{\mathrm{3}} +\mathrm{6}{h}^{\mathrm{2}} {p}^{\mathrm{2}} +\mathrm{4}{h}^{\mathrm{3}} {p}+{h}^{\mathrm{4}} \\ $$$$+{as}^{\mathrm{2}} \left({p}^{\mathrm{2}} +\mathrm{2}{hp}+{h}^{\mathrm{2}} \right) \\ $$$$+{bs}^{\mathrm{3}} \left({p}+{h}\right)+{cs}^{\mathrm{4}} =\mathrm{0} \\ $$$$\Rightarrow \\ $$$${p}^{\mathrm{4}} +\mathrm{4}{hp}^{\mathrm{3}} +\left(\mathrm{6}{h}^{\mathrm{2}} +{as}^{\mathrm{2}} \right){p}^{\mathrm{2}} \\ $$$$+\left(\mathrm{4}{h}^{\mathrm{3}} +\mathrm{2}{ahs}^{\mathrm{2}} +{bs}^{\mathrm{3}} \right){p} \\ $$$$+\left({h}^{\mathrm{4}} +{as}^{\mathrm{2}} {h}^{\mathrm{2}} +{bs}^{\mathrm{3}} {h}+{cs}^{\mathrm{4}} \right)=\mathrm{0} \\ $$$${let}\:\:\left(\sqrt{\mathrm{2}}{p}^{\mathrm{2}} +{Ap}+{B}\right)^{\mathrm{2}} =\left({p}^{\mathrm{2}} +{mp}+{k}\right)^{\mathrm{2}} \\ $$$$\Rightarrow\:\mathrm{2}\sqrt{\mathrm{2}}{A}−\mathrm{2}{m}=\mathrm{4}{h} \\ $$$$\mathrm{2}\sqrt{\mathrm{2}}{B}+{A}^{\mathrm{2}} −{m}^{\mathrm{2}} −\mathrm{2}{k} \\ $$$$\:\:\:\:\:\:\:=\:\mathrm{6}{h}^{\mathrm{2}} +{as}^{\mathrm{2}} \\ $$$$\mathrm{2}{AB}−\mathrm{2}{mk}=\mathrm{4}{h}^{\mathrm{3}} +\mathrm{2}{ah}+{bs}^{\mathrm{3}} \\ $$$${B}^{\mathrm{2}} −{k}^{\mathrm{2}} ={h}^{\mathrm{4}} +{as}^{\mathrm{2}} {h}^{\mathrm{2}} +{bs}^{\mathrm{3}} {h}+{cs}^{\mathrm{4}} \\ $$$${let}\:\:{m}=\mathrm{0}\:\:\Rightarrow \\ $$$${A}=\sqrt{\mathrm{2}}{h} \\ $$$$\mathrm{2}\sqrt{\mathrm{2}}{B}=\mathrm{4}{h}^{\mathrm{2}} +{as}^{\mathrm{2}} +\mathrm{2}{k} \\ $$$$\mathrm{4}{h}^{\mathrm{3}} +{ahs}^{\mathrm{2}} +\mathrm{2}{hk}=\mathrm{4}{h}^{\mathrm{3}} +\mathrm{2}{ah}+{bs}^{\mathrm{3}} \\ $$$$\Rightarrow\:\left({as}^{\mathrm{2}} +\mathrm{2}{k}−\mathrm{2}{a}\right){h}={bs}^{\mathrm{3}} \\ $$$$\left(\mathrm{4}{h}^{\mathrm{3}} +{as}^{\mathrm{2}} +\mathrm{2}{k}\right)^{\mathrm{2}} \\ $$$$\:\:−\mathrm{8}{k}^{\mathrm{2}} =\mathrm{8}\left({h}^{\mathrm{4}} +{as}^{\mathrm{2}} {h}^{\mathrm{2}} +{bs}^{\mathrm{3}} {h}+{cs}^{\mathrm{4}} \right) \\ $$$${let}\:\:{k}={a} \\ $$$$\Rightarrow\:\:{ah}={bs} \\ $$$$\Rightarrow\:\left(\frac{\mathrm{4}{b}^{\mathrm{2}} {h}}{{a}^{\mathrm{2}} }+{a}+\frac{\mathrm{2}{a}^{\mathrm{3}} {h}^{\mathrm{2}} }{{b}^{\mathrm{2}} }\right)^{\mathrm{2}} −\frac{\mathrm{8}{b}^{\mathrm{4}} {h}^{\mathrm{4}} }{{a}^{\mathrm{2}} } \\ $$$$\:\:=\mathrm{8}\left(\frac{{b}^{\mathrm{4}} }{{a}^{\mathrm{4}} }+\frac{\mathrm{2}{b}^{\mathrm{2}} }{{a}}+{c}\right) \\ $$$$\Rightarrow\:\frac{\mathrm{16}{b}^{\mathrm{4}} {h}^{\mathrm{2}} }{{a}^{\mathrm{4}} }+\frac{\mathrm{4}{a}^{\mathrm{6}} {h}^{\mathrm{4}} }{{b}^{\mathrm{4}} } \\ $$$$+\frac{\mathrm{8}{b}^{\mathrm{2}} {h}}{{a}}+\mathrm{32}{ah}^{\mathrm{3}} +\frac{\mathrm{4}{a}^{\mathrm{4}} {h}^{\mathrm{2}} }{{b}^{\mathrm{2}} } \\ $$$$\:\:−\frac{\mathrm{8}{b}^{\mathrm{4}} {h}^{\mathrm{4}} }{{a}^{\mathrm{2}} }=\lambda \\ $$$$\Rightarrow\:\left(\frac{\mathrm{4}{a}^{\mathrm{6}} }{{b}^{\mathrm{4}} }−\frac{\mathrm{8}{b}^{\mathrm{4}} }{{a}^{\mathrm{2}} }\right){h}^{\mathrm{4}} +\mathrm{32}{ah}^{\mathrm{3}} \\ $$$$\:\:\:+\left(\frac{\mathrm{16}{b}^{\mathrm{4}} }{{a}^{\mathrm{4}} }+\frac{\mathrm{4}{a}^{\mathrm{4}} }{{b}^{\mathrm{2}} }\right){h}^{\mathrm{2}} +\frac{\mathrm{8}{b}^{\mathrm{2}} {h}}{{a}}−\lambda=\mathrm{0} \\ $$$$… \\ $$

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