Question

if (x+1/x)²=3,find the value of x³+1/x^3.

Answer 1

Dear student
$$\begin{gathered} {\left(x+\frac{1}{x}\right)}^2=3 \\ \Rightarrow x+\frac{1}{x}=\pm \sqrt{3} \\ Now,weknow \\ {\left(x+\frac{1}{x}\right)}^3=x^3+\frac{1}{x^3}+3x+3\frac{1}{x} \\ \Rightarrow {\left(x+\frac{1}{x}\right)}^3=x^3+\frac{1}{x^3}+3\left(x+\frac{1}{x}\right) \\ \Rightarrow x^3+\frac{1}{x^3}={\left(x+\frac{1}{x}\right)}^3-3\left(x+\frac{1}{x}\right) \\ Ifx+\frac{1}{x}=\sqrt{3}, \\ Then,x^3+\frac{1}{x^3}={\left(\sqrt{3}\right)}^3-3\sqrt{3} \\ \Rightarrow x^3+\frac{1}{x^3}=3\sqrt{3}-3\sqrt{3} \\ \Rightarrow x^3+\frac{1}{x^3}=0 \\ Ifx+\frac{1}{x}=-\sqrt{3}, \\ Then,x^3+\frac{1}{x^3}={\left(-\sqrt{3}\right)}^3-3\left(-\sqrt{3}\right) \\ \Rightarrow x^3+\frac{1}{x^3}=-3\sqrt{3}+3\sqrt{3} \\ \Rightarrow x^3+\frac{1}{x^3}=0 \end{gathered}$$
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