X^2+1/x^2+2 (x+1/x) = 142/9

Произведем замену (x + 1/x) = t. Тогда t^2 = (x + 1/x) ^2 = x^2 + 2 * x * 1/x + 1/x^2 = x^2 + 1/x^2 + 2, откуда x^2 + 1/x^2 = t^2 - 2.

Подставим в исходное уравнение и решим его:

t^2 - 2 + 2t = 142/9;

t^2 + 2t - 160/9 = 0;

D = 2^2 - 4 * 1 * (-160/9) = 676/9 = (26/3) ^2;

t_1 = (-2 - 26/3) / 2 = - 16/3;

t_2 = (-2 + 26/3) / 2 = 10/3.

Проведем обратную замену и рассмотрим оба случая:

1) x + 1/x = - 16/3;

x^2 + 16/3 * x + 1 = 0;

D = (16/3) ^2 - 4 * 1 * 1 = 220/9 = (2*sqrt (55) / 3) ^2;

x_1 = (-16/3 - 2*sqrt (55) / 3) / 2 = - (sqrt (55) + 8) / 3;

x_2 = (-16/3 + 2*sqrt (55) / 3) / 2 = - (sqrt (55) - 8) / 3;

2) x + 1/x = 10/3;

x^2 - 10/3 * x + 1 = 0;

D = (-10/3) ^2 - 4 * 1 * 1 = 64/9 = (8/3) ^2;

x_3 = (10/3 - 8/3) / 2 = 1/3;

x_4 = (10/3 + 8/3) / 2 = 3.

Ответ:

x_1 = - (sqrt (55) + 8) / 3;

x_2 = - (sqrt (55) - 8) / 3;

x_3 = (10/3 - 8/3) / 2 = 1/3;

x_4 = (10/3 + 8/3) / 2 = 3.