Решите уравнение: а) у³ – 6у = 0; б) 6х⁴ + 3,6х² = 0; в) х³ + 3х = 3,5х²; г) х³ – 0,1х = 0,3х²; д) 9х³ – 18х² – х + 2 = 0; е) у⁴ – у³ – 16у² + 16у = 0; ж) р³ – р² = р – 1; з) х⁴ – х² = Зх³ – 3х.

Решение:

а) у³ – 6у = 0

y (y² – 6) = 0

y = 0 или y² – 6 = 0

y² = 6

y = +-√6

y = 0; y = +-√6

б) 6х⁴ + 3,6х² = 0

6x² (x² + 0,6) = 0

x² = 0 или x² + 0,6 = 0

x = 0 x² = – 0,6 – нет корней

в) х³ + 3х = 3,5х²

x³ + 3x – 3,5x² = 0

x(x² + 3 – 3,5x) = 0

x = 0 или x² – 3,5x + 3 = 0

2x² – 7x + 6 = 0

D = (-7)² – 4 ⋅ 2 ⋅ 6 = 49 – 48 = 1 > 0

√D = √1 = 1

x₁ = -(-7) – 1 / 2 ⋅ 2 = 7 – 1 / 4 = 6 / 4 = 1,5

x₂ = -(-7) + 1 / 2 ⋅ 2 = 7 + 1 / 4 = 8 /4 = 2

x = 0; x = 1,5; x = 2

г) х³ – 0,1х = 0,3х²

x³ – 0,1x – 0,3x² = 0

x(x² – 0,1 – 0,3x) = 0

x = 0 или x² – 0,3x – 0,1 = 0

10x² – 3x – 1 = 0

D = (-3)² – 4 ⋅ 10 ⋅ (-1) = 9 + 40 = 49 > 0

√D = √49 = √7² = 7

x₁ = – (-3) – 7 / 2 ⋅ 10 = 3 – 7 / 20 = – 4 / 20 = -0,2

x₂ = – (-3) + 7 / 2 ⋅ 10 = 3 + 7 / 20 = 10/20 = 0,5

д) 9х³ – 18х² – х + 2 = 0

9x² (x – 2) – (x – 2) = 0

(x – 2) (9x² – 1) = 0

x – 2 = 0 или 9x² – 1 = 0

9x² = 1

x² = 1/9

x = +-√ 1/9

x = +- 1/3

x = 2; x = +-1/3

е) у⁴ – у³ – 16у² + 16у = 0

y³ (y – 1) – 16y(y – 1) = 0

(y – 1) (y³ – 16y) = 0

y(y – 1) (y² – 16) = 0

y = 0 или y – 1 = 0 или y² – 16 = 0

y = 1 y² = 16

y = +-√16

y = +-4

y = 0; y = 1; y = +-4

ж) р³ – р² = р – 1

p³ – p² – p + 1 = 0

p² (p – 1) – (p – 1) = 0

(p – 1) (p² – 1) = 0

p – 1 = 0 или p² – 1 = 0

p = 1 p² = 1

p = +-1

p = +-1

з) х⁴ – х² = 3х³ – 3х

x⁴ – x² – 3x³ + 3x = 0

x² (x² – 1) – 3x (x² – 1) = 0

(x² – 1) (x – 3) = 0

x (x² – 1) ( x – 3) = 0

x = 0 или x² – 1 = 0 или x – 3 = 0

x² = 1 x = 3

x = +-1

x = +- 1; x = 0; x = 3

Источник : Учебник по Алгебре 9 класса авторов Ю.Н. Макарычев, Н.Г. Миндюк, К.И. Нешков, С.Б. Суворова, 2014г.