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x-3-x-c-0-let-x-t-h-t-s-t-3-3ht-2-3h-2-1-t-h-3-h-c-0-let-t-p-q-p-q-s-p-3-q-3-3pq-p-q-3h-p-2-q-2-6hpq-3h-2-1-p-q-h-3-h-c-0-p-4-q-4-pq-p-2-q-

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Question Number 143582 by ajfour last updated on 16/Jun/21
x^3 −x−c=0 let x=t+h (t−s){t^3 +3ht^2 +(3h^2 −1)t +(h^3 −h−c)}=0 let t=p+q (p+q−s){p^3 +q^3 +3pq(p+q) +3h(p^2 +q^2 )+6hpq +(3h^2 −1)(p+q)+(h^3 −h−c)} = 0 p^4 +q^4 +pq(p^2 +q^2 )+3pq(p+q)^2 +3h(p^2 +q^2 )(p+q)+6hpq(p+q) +(3h^2 −1)(p+q)^2 +(h^3 −h−c)(p+q) −s(p^3 +q^3 )−3pqs(p+q) −3hs(p^2 +q^2 )−6hspq +(3h^2 −1)s(p+q) −s(h^3 −h−c)=0 let p^4 +q^4 =p^2 +q^2 (p^2 +q^2 )^2 −(p^2 +q^2 )−2p^2 q^2 =0 ⇒ p^2 +q^2 =(1/2)±(√((1/4)+2p^2 q^2 )) say z=(1/2)+(√((1/4)+2m^2 )) (p+q)^2 =z+2m ⇒z(1+m+3m+3h^2 −1−3hs) +(z+2m)^(1/2) {3hz+6mh +h^3 −h−c−3ms −s(z−m)+(3h^2 −1)s} +6m^2 +2(3h^2 −1)m−6hsm −s(h^3 −h−c)=0 let (3h−s)z+m(6h−2s) +h^3 −h−c+3sh^2 −s=0 & 4m+3h(h−s)=0 ⇒ 2(3h−s)z−3h(h−s)(3h−s) +3sh^2 −s+h^3 −h−c=0 let s=−3h ⇒ 12hz−80h^3 −h−c=0 & m=−3h^2 ⇒ 54h^4 −6h^2 (3h^2 −1) −54h^4 +3h(h^3 −h−c)=0 ⇒ 15h^4 −3h^2 +3ch=0 And if h≠0 ⇒ 5h^3 −h+c=0 D=(c^2 /(100))−((1/(15)))^3 ⇒ D≥0 if c≥ (2/(3(√(15)))) m=−3h^2 ; s=−3h; And 12hz−80h^3 −h−c=0 ⇒ z=((20h^2 )/3)+(1/(12))+(c/(12h)) p+q=(√(z+2m)) =(√(((2h^2 )/3)+((c+h)/(12h)))) x=t+h=p+q+h x=h+(√(((2h^2 )/3)+((c+h)/(12h)))) but (c/(12h))>−(((2h^2 )/3)+(1/(12))) ⇒ c<−8h^3 −h ⇒ 8h^3 +h+c<0 5h^3 −h+c=0 ⇒ 13h^3 +2c<0 ......
x3−x−c=0letx=t+h(t−s){t3+3ht2+(3h2−1)t+(h3−h−c)}=0lett=p+q(p+q−s){p3+q3+3pq(p+q)+3h(p2+q2)+6hpq+(3h2−1)(p+q)+(h3−h−c)}=0p4+q4+pq(p2+q2)+3pq(p+q)2+3h(p2+q2)(p+q)+6hpq(p+q)+(3h2−1)(p+q)2+(h3−h−c)(p+q)−s(p3+q3)−3pqs(p+q)−3hs(p2+q2)−6hspq+(3h2−1)s(p+q)−s(h3−h−c)=0letp4+q4=p2+q2(p2+q2)2−(p2+q2)−2p2q2=0⇒p2+q2=12±14+2p2q2sayz=12+14+2m2(p+q)2=z+2m⇒z(1+m+3m+3h2−1−3hs)+(z+2m)1/2{3hz+6mh+h3−h−c−3ms−s(z−m)+(3h2−1)s}+6m2+2(3h2−1)m−6hsm−s(h3−h−c)=0let(3h−s)z+m(6h−2s)+h3−h−c+3sh2−s=0&4m+3h(h−s)=0⇒2(3h−s)z−3h(h−s)(3h−s)+3sh2−s+h3−h−c=0lets=−3h⇒12hz−80h3−h−c=0&m=−3h2⇒54h4−6h2(3h2−1)−54h4+3h(h3−h−c)=0⇒15h4−3h2+3ch=0Andifh≠0⇒5h3−h+c=0D=c2100−(115)3⇒D⩾0ifc⩾2315m=−3h2;s=−3h;And12hz−80h3−h−c=0⇒z=20h23+112+c12hp+q=z+2m=2h23+c+h12hx=t+h=p+q+hx=h+2h23+c+h12hbutc12h>−(2h23+112)⇒c<−8h3−h⇒8h3+h+c<05h3−h+c=0⇒13h3+2c<0……

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