1) (x+1)(x+7)*(x+3)(x+5) = - 16
(x^2+8x+7)(x^2+8x+15) = - 16
Замена 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 части 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