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Question Number 124191 by Algoritm last updated on 01/Dec/20

Answered by MJS_new last updated on 01/Dec/20

$$\mathrm{well}\mathrm{just}\mathrm{solve}\mathrm{it}!$$$$x=\frac{1}{6}(1-\sqrt[3]{359-12\sqrt{78}}-\sqrt[3]{359+12\sqrt{78}}$$$$x\approx -2.17868$$

Answered by Dwaipayan Shikari last updated on 01/Dec/20

$$x^2(1+\frac{x}{x-1})=8$$$$\Rightarrow x^2(x-1)+x^3=8(x-1)\Rightarrow 2x^3-x^2-8x+8=0$$$$x=y+\frac{1}{6}$$$$2y^3+\frac{1}{108}+y(y+\frac{1}{6})-y^2-\frac{1}{36}-\frac{y}{3}-8y-\frac{4}{3}+8=0$$$$\Rightarrow y^3-\frac{49y}{12}+\frac{10}{3}-\frac{1}{108}=0\Rightarrow y^3-\frac{49}{12}y+\frac{359}{108}=0$$$$y=\sqrt[3]{-\frac{359}{216}+\sqrt{{\left(\frac{359}{216}\right)}^2+{\left(\frac{49}{36}\right)}^3}}-\sqrt[3]{-\frac{359}{216}-\sqrt{{\left(\frac{359}{216}\right)}^2+{\left(\frac{49}{36}\right)}^3}}$$$$x=\frac{1}{6}(1+\sqrt[3]{-359+\sqrt{{359}^2+{49}^3}}-\sqrt[3]{-359-\sqrt{{359}^2+{49}^3}})$$$$x=\frac{1}{6}(1+\sqrt[3]{-359+\sqrt{128881+2401.49}}-\sqrt[3]{-359-\sqrt{128881+2401.49}})$$$$x=\frac{1}{6}(1+\sqrt[3]{-359+\sqrt{246530}}-\sqrt[3]{-359-\sqrt{246530}})$$

Commented by Algoritm last updated on 01/Dec/20

$$\mathrm{thank}\mathrm{you}$$

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