Question Number 50278 by LYCON TRIX last updated on 15/Dec/18

Commented by ajfour last updated on 15/Dec/18

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Commented by maxmathsup by imad last updated on 15/Dec/18

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Commented by ajfour last updated on 15/Mar/25

Commented by ajfour last updated on 15/Mar/25

Answered by behi83417@gmail.com last updated on 17/Dec/18
![a+x=t^2 ,a−x=s^2 ⇒x=((t^2 −s^2 )/2),a=((t^2 +s^2 )/2) (√(a+s))+(√(a−t))=t^2 −s^2 a+s+a−t+2(√((a+s)(a−t)))=(t^2 −s^2 )^2 2(√((a+s)(a−t)))=(t^2 −s^2 )^2 +t−s−2a= =t^4 +s^4 −2t^2 s^2 −t^2 −s^2 +t−s ⇒4(a^2 −at+as−st)= =t^8 +s^8 +4t^4 s^4 +t^4 +s^4 +t^2 +s^2 + +2t^4 s^4 −4t^6 s^2 −2t^6 −2t^4 s^2 +2t^5 −2t^4 s −4t^2 s^6 −2s^4 t^2 −2s^6 +2ts^4 −2s^5 +4t^4 s^2 +4t^2 s^4 −4t^3 s^2 +4t^2 s^3 +2t^2 s^2 −2t^3 +2t^2 s −2s^2 t+2s^3 −2ts LHS=(t^2 +s^2 )^2 −2(t^2 +s^2 )(t−s)−4st= =t^4 +s^4 +2t^2 s^2 −2t^3 +2t^2 s−2s^2 t+2s^3 −4st RHS=t^8 +s^8 +6t^4 s^4 +t^4 +s^4 +t^2 +s^2 −4t^6 s^2 −2t^6 −2s^6 +2t^4 s^2 +2t^5 −2t^4 s −4t^2 s^6 −2s^4 t^2 +2ts^4 −2s^5 +4t^2 s^4 −4t^3 s^2 +4t^2 s^3 +2t^2 s^2 −2t^3 +2t^2 s−2s^2 t+2s^3 −2ts ⇒t^8 +s^8 −2t^6 −2s^6 +2t^5 +2s^5 +t^2 +s^2 +2ts−4t^6 s^2 +2t^4 s^2 −2t^4 s−4t^2 s^6 +2s^4 t^2 +2ts^4 −4t^3 s^2 +4t^2 s^3 +2t^2 s −2s^2 t=0 (t^8 +s^8 )−2(t^6 +s^6 )+2(t^5 −s^5 )− −4t^2 s^2 (t^4 +s^4 )−2ts(t^3 −s^3 )+ +(2t^2 s^2 +1)(t^2 +s^2 )−4t^2 s^2 (t−s)+ +6t^4 s^4 +2ts=0 t−s=p,ts=q t^8 +s^8 =(t^4 −s^4 )^2 +2t^4 s^4 = =(t^2 −s^2 )^2 (t^2 +s^2 )^2 +2t^4 s^4 = =(t−s)^2 (t+s)^2 (t^2 +s^2 )^2 +2t^4 s^4 = =(t−s)^2 ((t−s)^2 +4ts)((t−s)^2 +2ts)^2 +2t^4 s^4 = =(p)^2 (p^2 +4q)(p^4 +4p^2 q+4q^2 )+2q^4 = =(p^4 +4p^2 q)(p^4 +4p^2 q+4q^2 )+2q^4 = =p^8 +8p^6 q+20p^4 q^2 +16p^2 q^3 +2q^4 t^6 +s^6 =(t^3 −s^3 )^2 +2t^3 s^3 =(t−s)^2 (t^2 +ts+s^2 )^2 +2t^3 s^3 = =(p)^2 ((t−s)^2 +3ts)^2 +2t^3 s^3 = =p^2 (p^2 +3q)^2 +2q^3 = =p^6 +6p^4 q+9p^2 q^2 +2q^3 t^5 −s^5 =(t−s)(t^4 +t^3 s+t^2 s^2 +ts^3 +s^4 )= =(t−s)(t^4 +s^4 +ts(t^2 +ts+s^2 ))= =(t−s)((t^2 +s^2 )^2 −2t^2 s^2 +ts((t−s)^2 +3ts)= =(t−s)[((t−s)^2 +2ts)^2 −2t^2 s^2 +ts((t−s)^2 +3ts))= =p[((p^2 +2q)^2 −2q^2 +q(p^2 +3q))= =p[(p^4 +4p^2 q+4q^2 )−2q^2 +qp^2 +3pq]= =p^5 +5p^2 q+5pq^2 t^4 +s^4 =(t^2 +s^2 )^2 −2t^2 s^2 =[(t−s)^2 +2ts]^2 −2t^2 s^2 = =[p^2 +2q]^2 −2q^2 =p^4 +4p^2 q+2q^2 t^3 −s^3 =(t−s)(t^2 +s^2 +ts)=p^3 +3pq t^2 +s^2 =(t−s)^2 +2ts=p^2 +2q after replacing and simplifing: (p^8 −2p^6 +2p^5 +p^2 )+16p^2 (p^2 −1)q^2 +(8p^6 −12p^4 +8p^3 +4)q=0](https://www.tinkutara.com/question/Q50305.png)
Commented by behi83417@gmail.com last updated on 17/Dec/18
