3 (х²+1/х²) + 5 (х+1/х) = 2

Найдем корень уравнения.

3 * (х² + 1/х²) + 5 * (х + 1/х) = 2;

Пусть x + 1/x = a, тогда:

a² = (x + 1/x) ²;

a² = x² + 2 * x * 1/x + 1/x²;

a² = x² + 2 + 1/x²;

x² + 1/x² = a² - 2;

Получаем:

3 * (a² - 2) + 5 * a = 2;

3 * a² - 3 * 2 + 5 * a - 2 = 0;

3 * a² - 6 + 5 * a - 2 = 0;

3 * a² + 5 * a - 8 = 0;

D = b² - 4 * a * c = 5² - 4 * 3 * (-8) = 25 + 12 * 8 = 25 + 80 + 16 = 41 + 80 = 121 = 11²;

a1 = (-5 + 11) / (2 * 3) = 6/6 = 1;

a2 = (-5 - 11) / 6 = - 16/6 = - 8/3;

Отсюда:

1) x + 1/x = 1;

x * x + 1/x * x = 1 * x;

x^2 - x + 1 = 0;

D = 1 - 4 * 1 * 1 = - 3;

Нет корней.

2) x + 1/x = - 8/3;

x * x + 1/x * x = - 8/3 * x;

x^2 + 1 = - 8/3 * x;

x^2 * 3 + 1 * 3 = - 8 * x;

3 * x^2 + 8 * x + 3 = 0;

D = 64 - 4 * 3 * 3 = 64 - 4 * 9 = 64 - 36 = 25 = 5²;

x1 = (-8 + 5) / 6 = - 3/6 = - 0.5;

x2 = (-8 - 5) / 6 = - 13/6 = - 2 1/6;

Ответ: х = - 0,5 и х = - 2 1/6.