Как решить уравнение? [latex]x^{2}+ \frac{1}{x^{2}} +x+ \frac{1}{x}=0 [/latex]

[latex]x^2+\frac{1}{x^2}+x+\frac{1}{x}=0\\\\t=x+\frac{1}{x}\; ;\; t^2=(x+\frac{1}{x})^2=x^2+\frac{1}{x^2}+2\; \; \to \; \; x^2+\frac{1}{x^2}=t^2-2\\\\t^2+t-2=0\\\\t_1=1,\; t_2=-2\; \; \to \\\\1)\; \; x+\frac{1}{x}=1\\\\x+\frac{1}{x}-1=0\; \; \to \; \; \frac{x^2-x+1}{x}=\; \; \to \; \; \left \{ {{x^2-x+1=0} \atop {x\ne 0}} \right. \\\\x^2-x+1=0\; ;\; D=1-4=-3\ \textless \ 0\; \to \; net\; reshenij\\\\2)\; \; x+\frac{1}{x}=-2[/latex] [latex]x+\frac{1}{x}+2=0\; \; \to \; \; \frac{x^2+2x+1}{x}=0\; \; \to \; \; \left \{ {{x^2+2x+1=0} \atop {x\ne 0}} \right. \\\\x^2+2x+1=(x+1)^2=0\; \; \to \; \; x=-1\\\\Otvet:\; x=-1.[/latex] [latex]P.S.(x+\frac{1}{x})^2=x^2+2*x*\frac{1}{x}+(\frac{1}{x})^2=x^2+2*1+\frac{1^2}{x^2}=\\\\=x^2+2+\frac{1}{x^2}[/latex]