6x² + 6/x² + 5x + 5/x - 38 = 0
6(x² + 1/x²) + 5(1/x + x) - 38 = 0
x ≠ 0
замена
1/x + x = t
(1/x + x)² = t²
1/x² + 2*1/x * x + x² = t²
1/x² + 2 + x² = t²
1/x² + x² = t² - 2
6(x² + 1/x²) + 5(1/x + x) - 38 = 0
6(t² - 2) + 5t - 38 = 0
6t² - 12 + 5t - 38 = 0
6t² + 5t - 50 = 0
D = 25 + 4*50*6 = 1225 = 35²
t12 = (-5 +- 35)/12 = 30/12 (5/2) - 40/12 (-10/3)
обратно к х
1. 1/x + x = 5/2
2x² - 5x + 2 = 0
D = 25 - 16 = 9 = 3²
x12 = (5 +- 3)/4 = 2 1/2
2. 1/x + x = -10/3
3x² + 10x + 3 = 0
D = 100 - 36 = 64 = 8²
x12 = (-10 +- 8)/6 = -3 -1/3
ответ x = {2,1/2,-3,-1/3}
вкратце
б) с³+d³-3cd(c+d) = (c+d)(с²-сd+d²)-3cd(c+d) = (c+d)((c²-cd+d²)-3cd) =
= (c+d)(c²-cd+d²-3cd) = (c+d)(c²-4cd+d²)
2. Пусть х - любое число, 2х - четное, 2х+1 - нечетное, 2х+3 - следующее нечетное. Тогда:
(2х+1)²-(2х+3)² = ((2х+1)-(2х+3))((2х+1)+(2х+3)) = (2х+1-2х-3)(2х+1+2х+3) =
= -2(4х+4) = -2*4(х+1) = -8(х+1)
-8(х+1) : 8 = -(х+1) чтд
3. 14⁴-165²+138²-107² = (196²-165²)+(138²-107²) =
= (196-165)(196+165)+(138-107)(138+107) = 31(196+165)+31(138+107) =
= 31((196+165)+(138+107))
31((196+165)+(138+107)) : 31 = ((196+165)+(138+107)) чтд