ОДЗ:
x ≠ 0
2(x² + 1/x²) - 7(x + 1/x) + 9 = 0
2(x² + 2 + 1/x² - 2) - 7(x + 1/x) + 9 = 0
2(x² + 2 + 1/x²) - 7(x + 1/x) + 5 = 0
2(x + 1/x)² - 7(x + 1/x) + 5 = 0
Пусть t = x + 1/x.
2t² - 7t + 5 = 0
2t² - 2t - 5t + 5 = 0
2t(t - 1) - 5(t - 1) = 0
(2t - 5)(t - 1) = 0
t = 1; 2,5
Обратная замена:
1) t = 1:
x + 1/x = 1
x² + 1 = x
x² - x + 1 = 0
D = 1 - 4 < 0 ⇒ нет корней
2) t = 5/2
x + 1/x = 5/2
2x² + 2 = 5x
2x² - 5x + 2 = 0
2x² - 4x - x + 2 = 0
2x(x - 2) - (x - 2) = 0
(2x - 1)(x - 2) = 0
x = 1/2; 2.
Ответ: x = 0,5; 2.