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x-3-x-c-0-solve-for-x-even-if-0-lt-c-lt-2-3-3-




Question Number 129687 by ajfour last updated on 17/Jan/21
    x^3 −x−c=0  (solve for x  even if   0<c<(2/(3(√3))))
$$\:\:\:\:{x}^{\mathrm{3}} −{x}−{c}=\mathrm{0} \\ $$$$\left({solve}\:{for}\:{x}\:\:{even}\:{if}\:\:\:\mathrm{0}<{c}<\frac{\mathrm{2}}{\mathrm{3}\sqrt{\mathrm{3}}}\right) \\ $$
Answered by ajfour last updated on 18/Jan/21
Just now I developed a new  approximate formula:  x=−((c/2))^(1/3) +(√(5((c/2))^(2/3) +(4/3)))   example:  y=(X−7)(X+2)(X+5)    =X^3 −39X−70  y=0  ⇒  X^3 −39X−70=0  let  x=(X/( (√(39))))  ⇒  x^3 −x−((70)/(39(√(39)))) =0   here  c=((70)/(39(√(39))))  x=−(((35)^(1/3) )/( (√(39))))+(√(5×(((35)^(2/3) )/(39))+(4/3)))  X=(√(39))x  X=−(35)^(1/3) +(√(5(35)^(2/3) +((4×39)/3)))  X=7.00022  (Eureka !)
$${Just}\:{now}\:{I}\:{developed}\:{a}\:{new} \\ $$$${approximate}\:{formula}: \\ $$$${x}=−\left(\frac{{c}}{\mathrm{2}}\right)^{\mathrm{1}/\mathrm{3}} +\sqrt{\mathrm{5}\left(\frac{{c}}{\mathrm{2}}\right)^{\mathrm{2}/\mathrm{3}} +\frac{\mathrm{4}}{\mathrm{3}}}\: \\ $$$${example}: \\ $$$${y}=\left({X}−\mathrm{7}\right)\left({X}+\mathrm{2}\right)\left({X}+\mathrm{5}\right) \\ $$$$\:\:={X}^{\mathrm{3}} −\mathrm{39}{X}−\mathrm{70} \\ $$$${y}=\mathrm{0}\:\:\Rightarrow\:\:{X}^{\mathrm{3}} −\mathrm{39}{X}−\mathrm{70}=\mathrm{0} \\ $$$${let}\:\:{x}=\frac{{X}}{\:\sqrt{\mathrm{39}}} \\ $$$$\Rightarrow\:\:{x}^{\mathrm{3}} −{x}−\frac{\mathrm{70}}{\mathrm{39}\sqrt{\mathrm{39}}}\:=\mathrm{0} \\ $$$$\:{here}\:\:{c}=\frac{\mathrm{70}}{\mathrm{39}\sqrt{\mathrm{39}}} \\ $$$${x}=−\frac{\left(\mathrm{35}\right)^{\mathrm{1}/\mathrm{3}} }{\:\sqrt{\mathrm{39}}}+\sqrt{\mathrm{5}×\frac{\left(\mathrm{35}\right)^{\mathrm{2}/\mathrm{3}} }{\mathrm{39}}+\frac{\mathrm{4}}{\mathrm{3}}} \\ $$$${X}=\sqrt{\mathrm{39}}{x} \\ $$$${X}=−\left(\mathrm{35}\right)^{\mathrm{1}/\mathrm{3}} +\sqrt{\mathrm{5}\left(\mathrm{35}\right)^{\mathrm{2}/\mathrm{3}} +\frac{\mathrm{4}×\mathrm{39}}{\mathrm{3}}} \\ $$$${X}=\mathrm{7}.\mathrm{00022} \\ $$$$\left(\mathcal{E}{ureka}\:!\right) \\ $$
Commented by Dwaipayan Shikari last updated on 18/Jan/21
How did you derive it sir?
$${How}\:{did}\:{you}\:{derive}\:{it}\:{sir}? \\ $$

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