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 ......


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Question Number 143582 by ajfour last updated on 16/Jun/21
$x^3 −x−c=0 let x=t+h (t−s){t^3 +3ht^2 +(3h^2 −1)t +(h^3 −h−c)}=0 let t=p+q (p+q−s){p^3 +q^3 +3pq(p+q) +3h(p^2 +q^2 )+6hpq +(3h^2 −1)(p+q)+(h^3 −h−c)} = 0 p^4 +q^4 +pq(p^2 +q^2 )+3pq(p+q)^2 +3h(p^2 +q^2 )(p+q)+6hpq(p+q) +(3h^2 −1)(p+q)^2 +(h^3 −h−c)(p+q) −s(p^3 +q^3 )−3pqs(p+q) −3hs(p^2 +q^2 )−6hspq +(3h^2 −1)s(p+q) −s(h^3 −h−c)=0 let p^4 +q^4 =p^2 +q^2 (p^2 +q^2 )^2 −(p^2 +q^2 )−2p^2 q^2 =0 ⇒ p^2 +q^2 =(1/2)±(√((1/4)+2p^2 q^2 )) say z=(1/2)+(√((1/4)+2m^2 )) (p+q)^2 =z+2m ⇒z(1+m+3m+3h^2 −1−3hs) +(z+2m)^(1/2) {3hz+6mh +h^3 −h−c−3ms −s(z−m)+(3h^2 −1)s} +6m^2 +2(3h^2 −1)m−6hsm −s(h^3 −h−c)=0 let (3h−s)z+m(6h−2s) +h^3 −h−c+3sh^2 −s=0 & 4m+3h(h−s)=0 ⇒ 2(3h−s)z−3h(h−s)(3h−s) +3sh^2 −s+h^3 −h−c=0 let s=−3h ⇒ 12hz−80h^3 −h−c=0 & m=−3h^2 ⇒ 54h^4 −6h^2 (3h^2 −1) −54h^4 +3h(h^3 −h−c)=0 ⇒ 15h^4 −3h^2 +3ch=0 And if h≠0 ⇒ 5h^3 −h+c=0 D=(c^2 /(100))−((1/(15)))^3 ⇒ D≥0 if c≥ (2/(3(√(15)))) m=−3h^2 ; s=−3h; And 12hz−80h^3 −h−c=0 ⇒ z=((20h^2 )/3)+(1/(12))+(c/(12h)) p+q=(√(z+2m)) =(√(((2h^2 )/3)+((c+h)/(12h)))) x=t+h=p+q+h x=h+(√(((2h^2 )/3)+((c+h)/(12h)))) but (c/(12h))>−(((2h^2 )/3)+(1/(12))) ⇒ c<−8h^3 −h ⇒ 8h^3 +h+c<0 5h^3 −h+c=0 ⇒ 13h^3 +2c<0 ......$
$x^{3} - x - c = 0$ $l e t x = t + h$ $(t - s) {t^{3} + 3 {h t}^{2} + (3 h^{2} - 1) t$ $+ (h^{3} - h - c)} = 0$ $l e t t = p + q$ $(p + q - s) {p^{3} + q^{3} + 3 p q (p + q)$ $+ 3 h (p^{2} + q^{2}) + 6 h p q$ $+ (3 h^{2} - 1) (p + q) + (h^{3} - h - c)}$ $= 0$ $p^{4} + q^{4} + p q (p^{2} + q^{2}) + 3 p q {(p + q)}^{2}$ $+ 3 h (p^{2} + q^{2}) (p + q) + 6 h p q (p + q)$ $+ (3 h^{2} - 1) {(p + q)}^{2}$ $+ (h^{3} - h - c) (p + q)$ $- s (p^{3} + q^{3}) - 3 p q s (p + q)$ $- 3 h s (p^{2} + q^{2}) - 6 h s p q$ $+ (3 h^{2} - 1) s (p + q)$ $- s (h^{3} - h - c) = 0$ $l e t p^{4} + q^{4} = p^{2} + q^{2}$ ${(p^{2} + q^{2})}^{2} - (p^{2} + q^{2}) - 2 p^{2} q^{2} = 0$ $\Rightarrow p^{2} + q^{2} = \frac{1}{2} \pm \sqrt{\frac{1}{4} + 2 p^{2} q^{2}}$ $s a y z = \frac{1}{2} + \sqrt{\frac{1}{4} + 2 m^{2}}$ ${(p + q)}^{2} = z + 2 m$ $\Rightarrow z (1 + m + 3 m + 3 h^{2} - 1 - 3 h s)$ $+ {(z + 2 m)}^{1 / 2} {3 h z + 6 m h$ $+ h^{3} - h - c - 3 m s$ $- s (z - m) + (3 h^{2} - 1) s}$ $+ 6 m^{2} + 2 (3 h^{2} - 1) m - 6 h s m$ $- s (h^{3} - h - c) = 0$ $l e t (3 h - s) z + m (6 h - 2 s)$ $+ h^{3} - h - c + 3 {s h}^{2} - s = 0$ $& 4 m + 3 h (h - s) = 0 \Rightarrow$ $2 (3 h - s) z - 3 h (h - s) (3 h - s)$ $+ 3 {s h}^{2} - s + h^{3} - h - c = 0$ $l e t s = - 3 h$ $\Rightarrow 12 h z - 80 h^{3} - h - c = 0$ $& m = - 3 h^{2}$ $\Rightarrow 54 h^{4} - 6 h^{2} (3 h^{2} - 1)$ $- 54 h^{4} + 3 h (h^{3} - h - c) = 0$ $\Rightarrow 15 h^{4} - 3 h^{2} + 3 c h = 0$ $A n d i f h \neq 0 \Rightarrow$ $5 h^{3} - h + c = 0$ $D = \frac{c^{2}}{100} - {(\frac{1}{15})}^{3}$ $\Rightarrow D ⩾ 0 i f c ⩾ \frac{2}{3 \sqrt{15}}$ $m = - 3 h^{2}; s = - 3 h; A n d$ $12 h z - 80 h^{3} - h - c = 0$ $\Rightarrow z = \frac{20 h^{2}}{3} + \frac{1}{12} + \frac{c}{12 h}$ $p + q = \sqrt{z + 2 m}$ $= \sqrt{\frac{2 h^{2}}{3} + \frac{c + h}{12 h}}$ $x = t + h = p + q + h$ $x = h + \sqrt{\frac{2 h^{2}}{3} + \frac{c + h}{12 h}}$ $b u t \frac{c}{12 h} > - (\frac{2 h^{2}}{3} + \frac{1}{12})$ $\Rightarrow c < - 8 h^{3} - h$ $\Rightarrow 8 h^{3} + h + c < 0$ $5 h^{3} - h + c = 0$ $\Rightarrow 13 h^{3} + 2 c < 0$ $. . . . . .$
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