Решите уравнение методом введения новой переменной

Объяснение:

3) (x^2 + 1/x^2) + 7(x - 1/x) + 10 = 0

Новая переменная y = x - 1/x, тогда

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

x^2 + 1/x^2 = y^2 + 2

Подставляем:

y^2 + 2 + 7y + 10 = 0

y^2 + 7y + 12 = 0

(y + 3)(y + 4) = 0

y1 = -3 = x - 1/x

x - 1/x + 3 = 0

x^2 + 3x - 1 = 0

D = 9 - 4(-1) = 13

x1 = (-3 - √13)/2; x2 = (-3 + √13)/2

y2 = -4 = x - 1/x

x - 1/x + 4 = 0

x^2 + 4x - 1 = 0

D = 16 - 4(-1) = 20

x3 = (-4 - √20)/2 = (-4 - 2√5)/2 = -2 - √5

x4 = (-4 + √20)/2 = (-4 + 2√5)/2 = -2 + √5

4) (x^2 + 4/x^2) - (x + 2/x) - 8 = 0

Замена y = x + 2/x, y^2 = x^2 + 2*x*2/x + 4/x^2

x^2 + 4/x^2 = y^2 - 4

y^2 - 4 - y - 8 = 0

y^2 - y - 12 = 0

(y - 4)(y + 3) = 0

y1 = -3 = x + 2/x

x + 2/x + 3 = 0

x^2 + 3x + 2 = 0

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

x1 = -1; x2 = -2

y2 = 4 = x + 2/x

x + 2/x - 4 = 0

x^2 - 4x + 2 = 0

D = 16 - 4*2 = 8

x3 = (4 - √8)/2 = (4 - 2√2)/2 = 2 - √2

x4 = (4 + √8)/2 = (4 + 2√2)/2 = 2 + √2