Решите уравнение: а) 0,7х4 – х3 = 0; б) 0,5х3 – 72х = 0; в) х3 + 4х = 5х2; г) Зх3 – х2 + 18х – 6 = 0; д) 2х4 – 18х2 = 5х3 – 45х; е) Зу2 -2у = 2у3 – 3.
Решение:
а) 0,7х4 – х3 = 0
x3 (0,7x – 1) = 0
x3 = 0 или 0,7x – 1 = 0
x = 0 0,7x = 1
x = 1 : 0,7
x = 1 3/7
x = 0; x = 1 3/7
б) 0,5х3 – 72х = 0
x (0,5x2 – 72) = 0
x = 0 или 0,5x2 – 72 = 0
0,5x2 = 72
x2 = 72 : 0,5
x2 = 144
x = +-√144
x = +-12
x = 0; x = +- 12
в) х3 + 4х = 5х2
x3 + 4x – 5x2 = 0
x(x2 – 5x + 4) = 0
x = 0 или x2 – 5x + 4 = 0
D = (-5)2 – 4 ⋅ 1 ⋅ 4 = 25 – 16 = 9 > 0
√D = √9 = √32 = 3
x1 = -(-5) – 3 / 2 = 5 – 3 / 2 = 2/2 = 1
x2 = -(-5) + 3 / 2 = 5 + 3 / 2 = 8/2 = 4
x = 0; x = 1; x = 4
г) Зх3 – х2 + 18х – 6 = 0;
x2 (3x – 1) + 6 (3x – 1) = 0
(3x – 1) ( x2 + 6) = 0
3x – 1 = 0 или x2 + 6 = 0
3x = 1 x2 = -6 нет корней
x = 1/3
д) 2х4 – 18х2 = 5х3 – 45х
2x4 – 18x2 – 5x3 + 45x = 0
2x2 (x2 – 9) – 5x(x2 – 9) = 0
(x2 – 9) (2x2 – 5x) = 0
x(x2 – 9) (2x – 5) = 0
x = 0 или x2 – 9 = 0 или 2x – 5 = 0
x2 = 9 2x = 5
x = +-√9 x = 5 :2
x = +-3 x = 2,5
x = 0; x = 2,5; x = +-3
е) Зу2 -2у = 2у3 – 3
3y2 – 2y – 2y3 + 3 = 0
3y2 + 3 – 2y – 2y3 = 0
3(y2 + 1) – 2y(1 + y2 ) = 0
(y2 + 1) ( 3 – 2y) = 0
y2 + 1 = 0 или 3 – 2y = 0
y2 = -1 корней нет 2y = 3
y = 3 : 2
y = 1,5
y = 1,5
Источник : Учебник по Алгебре 9 класса авторов Ю.Н. Макарычев, Н.Г. Миндюк, К.И. Нешков, С.Б. Суворова, 2014г.