42.20. Это так называемые возвратные уравнения.
Решаются заменой y = x + 1/x
1) 3x^2 + 5x + 5/x + 3/x^2 = 16
Замена x + 1/x = y, тогда
y^2 = (x+1/x)^2 = x^2 + 1/x^2 + 2x*1/x = x^2 + 1/x^2 + 2
x^2 + 1/x^2 = y^2 - 2
Получаем
3(y^2 - 2) + 5y - 16 = 0
3y^2 - 6 + 5y - 16 = 3y^2 + 5y - 22 = 0
(3y + 11)(y - 2) = 0
a) 3y + 11 = 0; y = x + 1/x = -11/3
3x^2 + 11x + 3 = 0
D = 11^2 - 4*3*3 = 121 - 36 = 75 = (5√3)^2
x1 = (-11 - 5√3)/6; x2 = (-11 + 5√3)/6
b) y = x + 1/x = 2
x^2 - 2x + 1 = 0
(x - 1)^2 = 0
x3 = x4 = 1
Ответ: x1 = (-11 - 5√3)/6; x2 = (-11 + 5√3)/6; x3 = 1
2) x^2 + 36/x^2 = 112/5*(x/2 - 3/x)
Замена x/2 - 3/x = y, тогда
y^2 = x^2/4 - 2*x/2*3/x + 9/x^2 = 1/4*(x^2 + 36/x^2) - 3
x^2 + 36/x^2 = 4y^2 + 12
Получаем
4y^2 + 12 = 112/5*y
20y^2 - 112y + 60 = 0
Делим все на 4
5y^2 - 28y + 15 = 0
D/4 = 14^2 - 5*15 = 196 - 75 = 121 = 11^2
a) y1 = x/2 - 3/x = (14 - 11)/5 = 3/5
5x^2 - 6x - 15 = 0
D/4 = 3^2 + 5*15 = 9 + 75 = 84 = (2√21)^2
x1 = (3 - 2√21)/5; x2 = (3 + 2√21)/5
b) y2 = x/2 - 3/x = (14 + 11)/5 = 5
x^2 - 10x - 6 = 0
D/4 = 5^2 + 6 = 25 + 6 = 31
x3 = 5 - √31; x4 = 5 + √31
Ответ: x1 = (3 - 2√21)/5; x2 = (3 + 2√21)/5; x3 = 5 - √31; x4 = 5 + √31
3) (x^2 + 1)^2 / [x(x+1)^2] = 625/112
112(x^4 + 2x^2 + 1) = 625x(x^2 + 2x + 1)
112x^4 + 224x^2 + 112 = 625x^3 + 1250x^2 + 625x
112x^4 - 625x^3 - 1026x^2 - 625x + 112 = 0
Делим все на x^2
112x^2 - 625x - 1026 - 625*1/x + 112/x^2 = 0
112*(x^2 + 1/x^2) - 625*(x + 1/x) - 1026 = 0
Замена x + 1/x = y. Тогда, как в 1) номере, x^2 + 1/x^2 = y^2 - 2
112(y^2 - 2) - 625y - 1026 = 0
112y^2 - 625y - 1250 = 0
D = 625^2 - 4*112*(-1250) = 625*(625 + 4*112*2) = 625*1521 = (25*39)^2
a) y1 = (625 - 25*39)/(2*112) = 25(25-39)/224 < 0
решений нет
b) y2 = (625 + 25*39)/(2*112) = 25(25+39)/224 = 25*64/224 = 50/7
x + 1/x = 50/7
7x^2 - 50x + 7 = 0
D/4 = 25^2 - 7*7 = 625 - 49 = 576 = 24^2
x1 = (25-24)/7 = 1/7; x2 = (25+24)/7 = 49/7 = 7
Ответ: x1 = 7; x2 = 1/7
4) 4x^4 - 8x^3 + 3x^2 - 8x + 4 = 0
Делим все на x^2
4x^2 - 8x + 3 - 8/x + 4/x^2 = 0
Замена x + 1/x = y. Тогда, как в 1) номере, x^2 + 1/x^2 = y^2 - 2
4(y^2 - 2) - 8y + 3 = 0
4y^2 - 8y - 5 = 0
D/4 = 4^2 + 4*5 = 36 = 6^2
a) y1 = (4 - 6)/4 = -2/4 = -1/2
x + 1/x = -1/2
2x^2 + x + 2 = 0
Решений нет
b) y2 = (4 + 6)/4 = 10/4 = 5/2
x + 1/x = 5/2
2x^2 - 5x + 2 = 0
(2x - 1)(x - 2) = 0
Ответ: x1 = 2; x2 = 1/2
42.22. Эти уравнения решаются делением и сведением к квадратному
1) x^4 + 5x^2(x+1) = 6(x+1)^2
Делим все на (x+1)^2
x^4/(x+1)^2 + 5x^2/(x+1) - 6 = 0
Замена x^2/(x+1) = y
y^2 + 5y - 6 = 0
(y + 6)(y - 1) = 0
a) y1 = x^2/(x+1) = -6
x^2 = -6(x + 1)
x^2 + 6x + 6 = 0
D/4 = 3^2 - 6 = 9 - 6 = 3
x1 = -3 - √3; x2 = -3 + √3
b) y2 = x^2/(x+1) = 1
x^2 = x + 1
x^2 - x - 1 = 0
D = 1 - 4(-1) = 5
x3 = (1 - √5)/2; x4 = (1 + √5)/2
Ответ: x1 = -3 - √3; x2 = -3 + √3; x3 = (1 - √5)/2; x4 = (1 + √5)/2
2) (x^2-3x+1)^2 + 3(x-1)(x^2-3x+1) = 4(x-1)^2
Делим все на (x-1)^2
(x^2-3x+1)^2/(x-1)^2 + 3(x^2-3x+1)/(x-1) - 4 = 0
Замена (x^2-3x+1)/(x-1) = y
y^2 + 3y - 4 = 0
(y - 1)(y + 4) = 0
a) y = (x^2-3x+1)/(x-1) = -4
x^2 - 3x + 1 = 4x - 4
x^2 - 7x + 5 = 0
D = 7^2 - 4*5 = 49 - 20 = 29
x1 = (7 - √29)/2; x2 = (7 + √29)/2
b) y = (x^2-3x+1)/(x-1) = 1
x^2 - 3x + 1 = x - 1
x^2 - 4x + 2 = 0
D/4 = 2^2 - 2 = 4 - 2 = 2
x3 = 4 - √2; x4 = 4 + √2
Ответ: x1 = (7 - √29)/2; x2 = (7 + √29)/2; x3 = 4 - √2; x4 = 4 + √2