Question

Решите уравнение:

$x^{2}+5x+frac{5}{x} +frac{1}{x^{2}}=12$

Answer 1

$x^2+5x+frac{5}{x}+frac{1}{x^2}=12\\ x^2+2+frac{1}{x^2}+5(x+frac{1}{x})=12+2\\ x^2+2*x*frac{1}{x}+(frac{1}{x})^2+5(x+frac{1}{x})=12+2\\ (x+frac{1}{x})^2+5*(x+frac{1}{x})-14=0\\ x+frac{1}{x}=t\\ t^2+5t-14=0\\ t^2+7t-2t-14=0\\ t(t+7)-2(t+7)=0\\ (t-2)(t+7)=0\\ t-2=0 or t+7=0\\ t=2 or t=-7\\ x+frac{1}{x}=2 or x+frac{1}{x}=-7\\ x^2-2x+1=0 or x^2+7x+1=0\\ (x-1)^2=0 or x_{1,2}=frac{-7pmsqrt{45}}{2}$

$x_3=1 or x_{1,2}=frac{-7pm3sqrt{5}}{2}$

Ответ: $1; frac{-7pm3sqrt{5}}{2}$