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Question Number 192883 by ajfour last updated on 30/May/23

	Answered by ajfour last updated on 30/May/23
	$(p+s)^3 −(p+s)=k p^3 −p=c s{(p+s)^2 +p^2 +p(p+s)}−s=k−c 3p^2 +3sp+s^2 =((s+k−c)/s) 3sp^2 =(((s+k−c−s^3 −3s)p)/s)−3c =s+k−c−s^3 −3s^2 p p=((s(k+s+2c−s^3 ))/(2s^3 −2s−c+k)) 3s^2 (−s^3 +s+2c+k)^2 + 3s^2 (−s^3 +s+2c+k)(2s^3 −2s−c+k) =(−s^3 +s−c+k)(2s^3 −2s−c+k)^2 if s^3 =(c/2) 3s^2 (s+((3c)/2)+k)^2 +3s^2 (s+((3c)/2)+k)(k−2s) =(s−((3c)/2)+k)(k−2s)^2 p=((s(k+s+2c−s^3 ))/(2s^3 −2s−c+k)) =((s(s+((3c)/2)+k))/(k+s−3s)) ⇒ p(k+s−3s)=s(k+s+((3c)/2)) k+s=((3sp+((3cs)/2))/(p−s))=((3s(p+(c/2)))/(p−s)) 3p^2 +3sp=s−((3c)/2)+k =((3s(p+(c/2))−((3cp)/2)+((3cs)/2))/(p−s)) 3p(p+s) =((3p(s−(c/2))+3cs)/(p−s)) p(p^2 −s^2 )=p(s−(c/2))+cs p^3 −{s(s+1)−(c/2)}p−cs=0 ((p/(2s)))^3 −{(1/4)[1+((2/c))^(1/3) ]−(1/8)}(p/(2s))−(1/4)((c/2))^(1/3) =0 ....$
	${(p + s)}^{3} - (p + s) = k$ $p^{3} - p = c$ $s {{(p + s)}^{2} + p^{2} + p (p + s)} - s = k - c$ $3 p^{2} + 3 s p + s^{2} = \frac{s + k - c}{s}$ $3 {s p}^{2} = \frac{(s + k - c - s^{3} - 3 s) p}{s} - 3 c$ $= s + k - c - s^{3} - 3 s^{2} p$ $p = \frac{s (k + s + 2 c - s^{3})}{2 s^{3} - 2 s - c + k}$ $3 s^{2} {(- s^{3} + s + 2 c + k)}^{2} +$ $3 s^{2} (- s^{3} + s + 2 c + k) (2 s^{3} - 2 s - c + k)$ $= (- s^{3} + s - c + k) {(2 s^{3} - 2 s - c + k)}^{2}$ $i f s^{3} = \frac{c}{2}$ $3 s^{2} {(s + \frac{3 c}{2} + k)}^{2} + 3 s^{2} (s + \frac{3 c}{2} + k) (k - 2 s)$ $= (s - \frac{3 c}{2} + k) {(k - 2 s)}^{2}$ $p = \frac{s (k + s + 2 c - s^{3})}{2 s^{3} - 2 s - c + k}$ $= \frac{s (s + \frac{3 c}{2} + k)}{k + s - 3 s}$ $\Rightarrow p (k + s - 3 s) = s (k + s + \frac{3 c}{2})$ $k + s = \frac{3 s p + \frac{3 c s}{2}}{p - s} = \frac{3 s (p + \frac{c}{2})}{p - s}$ $3 p^{2} + 3 s p = s - \frac{3 c}{2} + k$ $= \frac{3 s (p + \frac{c}{2}) - \frac{3 c p}{2} + \frac{3 c s}{2}}{p - s}$ $3 p (p + s) = \frac{3 p (s - \frac{c}{2}) + 3 c s}{p - s}$ $p (p^{2} - s^{2}) = p (s - \frac{c}{2}) + c s$ $p^{3} - {s (s + 1) - \frac{c}{2}} p - c s = 0$ ${(\frac{p}{2 s})}^{3} - {\frac{1}{4} [1 + {(\frac{2}{c})}^{1 / 3}] - \frac{1}{8}} \frac{p}{2 s} - \frac{1}{4} {(\frac{c}{2})}^{1 / 3} = 0$ $. . . .$
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