Алгебра даю 20 раутов решить (2не надобно)

1) (x+1)(x+7)*(x+3)(x+5) = - 16

(x^2+8x+7)(x^2+8x+15) = - 16

Подмена x^2+8x = y

(y+7)(y+15) = - 16

y^2 + 22y + 105+16 = 0

y^2 + 22y + 121 = 0

(y + 11)^2 = 0

y = x^2 + 8x = - 11

x^2 + 8x + 11 = 0

D/4 = 4^2 - 11 = 16 - 11 = 5

Ответ: x1 = - 4 - 5; x2 = -4 + 5

3) (x^2-x)/(x^2-x+1) - (x^2-x+2)/(x^2-x-2) = 1

Замена x^2-x = y

y/(y+1) - (y+2)/(y-2) = 1

Выделяем целые доли в каждой дроби

(y+1-1)/(y+1) - (y-2+4)/(y-2) = 1

1 - 1/(y+1) - 1 - 4/(y-2) = 1

-1/(y+1) - 4/(y-2) = 1

Приводим к общему знаменателю (y+1)(y-2) и умножаем на него.

-(y-2) - 4(y+1) - (y+1)(y-2) = 0

-y+2-4y-4-y^2+y+2 = 0

-y^2 - 4y = 0

Умножаем на -1

y^2 + 4y = 0

y(y + 4) = 0

y1 = x^2 - x = 0

x(x - 1) = 0

x1 = 0; x2 = 1

y2 = x^2 - x = -4

x^2 - x + 4 = 0

Корней нет.

Ответ: x1 = 0; x2 = 1

4) 3x^2 + 5x + 5/x + 3/x^2 = 16

3(x^2 + 1/x^2) + 5(x + 1/x) = 16

Замена x + 1/x = y, тогда y^2 = x^2+2x*1/x+1/x^2 = x^2+2+1/x^2

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

3(y^2 - 2) + 5y = 16

3y^2 - 6 + 5y - 16 = 0

3y^2 + 5y - 22 = 0

D = 5^2 - 4*3(-22) = 25 + 12*22 = 289 = 17^2

y1 = x + 1/x = (-5 - 17)/6 = -22/6 = -11/3

x + 1/x + 11/3 = 0

3x^2 + 11x + 3 = 0

D1 = 11^2 - 4*3*3 = 121 - 36 = 85

x1 = (-11 - 85)/6; x2 = (-11 + 85)/6

y2 = x + 1/x = (-5 + 17)/6 = 12/6 = 2

x + 1/x - 2 = 0

x^2 - 2x + 1 = 0

(x - 1)^2 = 0

x3 = x4 = 1

Ответ: x1 = (-11 - 85)/6; x2 = (-11 + 85)/6; x3 = x4 = 1

1)

(х+3)(х+1)(х+5)(х+7) = -16,

(х+1)(х+7) * (х+3)(х+5) = -16,

(х+х+7х+7) * (х+3х+5х+15) = -16,

(х+8х+7) * (х+8х+15) = -16,

пусть (х+8х+7) = а, тогда  (х+8х+15) = (а+8), означает:

а*(а+8) = -16,

а + 8а + 16 = 0,

Д = 8 - 4*1*16 = 64 - 64 = 0,

а = -8 / 2*1 = -8/2 = -4,

х + 8х + 7 = а,

х + 8х + 7 = -4,

х + 8х + 11 = 0,

Д = 8 - 4*1*11 = 64 - 44 = 20,

х1 = (-8 + 20) / 2*1 = -4 + 5,

х2 = (-8 - 20) / 2*1 = -(4 + 5),