2 (x²+1/x²) - 11 (x-1/x) + 8=0

1. Обозначим:

x - 1/x = y, отсюда: x^2 - 2 + 1/x^2 = y^2; x^2 + 1/x^2 = y^2 + 2; 2 (x^2 + 1/x^2) - 11 (x - 1/x) + 8 = 0; 2 (y^2 + 2) - 11y + 8 = 0; 2y^2 + 4 - 11y + 8 = 0; 2y^2 - 11y + 12 = 0; D = 11^2 - 4 * 2 * 12 = 121 - 96 = 25; y = (11 ± √25) / 4 = (11 ± 5) / 4;

a) y = (11 - 5) / 4 = 6/4 = 3/2;

b) y = (11 + 5) / 4 = 16/4 = 4.

2. Обратная замена:

x - 1/x = y; x^2 - 1 = yx; x^2 - yx - 1 = 0;

a) y = 3/2;

x^2 - 3/2 * x - 1 = 0; 2x^2 - 3x - 2 = 0; D = 3^2 + 4 * 2 * 2 = 25 = 5^2; x = (3 ± 5) / 4; x1 = (3 - 5) / 4 = - 2/4 = - 1/2; x2 = (3 + 5) / 4 = 8/4 = 2.

b) y = 4;

x^2 - 4x - 1 = 0; D/4 = 2^2 + 1 = 5; x = 2 ± √5.

Ответ: - 1/2; 2; 2 ± √5.