Решить уравнение:

(x^2 + 1/x^2) + (x + 1/x) = 0

(x + 1/x) ^2 = x^2 + 2 + 1/x^2

Замена: t = x + 1/x

(t^2 - 2) + t = 0

t^2 + t - 2 = 0

t1 = 1; t2 = - 2

x + 1/x = 1

x^2 - x + 1 = 0 решений нет

x + 1/x = - 2

x^2 + 2x + 1 = 0

x = - 1

(x^2 + 1/x^2) + (x + 1/x) = 0

ОДЗ x0

x+1/x=t

(x^2+1/x^2) = (x^2+2*x^2*1/x^2+1/x^2) - 2 = (1/x+x) ^2-2

t^2-2+t=0

D=1+8=9

t12 = (-1+-3) / 2=1 - 2

x+1/x=1

x^2-x+1=0

D=-3 решений нет

x+1/x=-2

x^2+2x+1=0

(x+1) ^2=0

x=-1