3(6).
а)
(6х+1)(6х-1)(12х-1)(4х+1) = 56,
(6х+1)(12х-1)*(6х-1)(4х+1) = 56,
(72х²+12х-6х-1)*(24х²+6х-4х-1) = 56,
(72х²+6х-1)*(24х²+2х-1) = 56,
(6*(12х²+х)-1)*(2*(12х²+х)-1) = 56,
пусть (12х²+х) = а, тогда:
(6а-1)(2а-1) = 56,
12а²-6а-2а+1 = 56,
12а²-8а+1-56 = 0,
12а²-8а-55 = 0,
Д = (-8)² - 4*12*(-55) = 64 + 2640 = 2704,
а1= (8 + √2704) / 2*12 = (8 + 52) / 24 = 60/24 = 5/2 = 2,5,
а2= (8 - √2704) / 2*12 = (8 - 52) / 24 = -44/24 = -11/6 = -1 5/6 < 0 - не подх,
при а = 2,5:
12х²+х = 2,5,
12х²+х-2,5 = 0,
Д = 1² - 4*12*(-2,5) = 1 + 120 = 121,
х1 = (-1 + √121) / 2*12 = (-1+11)/24 = 10/24 = 5/12,
х2 = (-1 - √121) / 2*12 = (-1-11)/24 = -12/24 = -1/2,
б)
(6х+1)(6х-1)(12х-1)(3х+1) = -18х²,
(6х+1)(6х-1)*(12х-1)(3х+1) = -18х²,
(36х²-1)*(36х²+12х-3х-1) = -18х²,
(36х²-1)*(36х²+9х-1) = -18х²,
(36х²-1)/х * (36х²+9х-1)/х = -18х²/х²,
(36х²/х - 1/х) * (36х²/х + 9х/х - 1/х) = -18х²/х²,
(36х - 1/х) * (36х + 9 - 1/х) = -18,
пусть (36х-1/х) = а, тогда:
а * (а + 9) = -18,
а² + 9а + 18 = 0,
Д = 9² - 4*1*18 = 81 - 72 = 9,
а1 = (-9 + √9) / 2*1 = (-9+3) / 2 = -6/2 = -3,
а2 = (-9 - √9) / 2*1 = (-9-3) / 2 = -12/2 = -6,
при а1 = -3:
36х - 1/х = -3,
36х² - 1 = -3х,
36х² + 3х - 1 = 0,
Д = 3² - 4*36*(-1) = 9 + 144 = 153,
х1 = (-3 + √153) / 2*36 = (-3 + 3√17) / 72 = (-1+√17)/24,
х2 = (-3 - √153) / 2*36 = (-3 - 3√17) / 72 = (-1-√17)/24,
при а2 = -6:
36х - 1/х = -6,
36х² - 1 = -6х,
36х² + 6х - 1 = 0,
Д = 6² - 4*36*(-1) = 36 + 144 = 180,
х3 = (-6 + √180) / 2*36 = (-6 + 6√5) / 72 = (-1+√5)/12,
х4 = (-6 - √180) / 2*36 = (-6 - 6√5) / 72 = (-1-√5)/12