Решите уравнение: а) х³ + 7х² – 6 = 0; б) х³ + 4х² – 5 = 0.

Решение:

а) х³ + 7х² – 6 = 0

x³ + 7x² – 6 = x³ + x² + 6x² – 6 = x² (x + 1) + 6(x² – 1) = x² (x + 1) + 6 (x – 1) ⋅ ( x + 1) = (x + 1) ⋅ (x² + 6(x – 1) = (x + 1) ⋅ ( x² + 6x – 6)

(x + 1) ⋅ (x² + 6x – 6) = 0

x + 1 = 0 или x² + 6x – 6 = 0

x = -1

D = 6² – 4 ⋅ 1 ⋅ (-6) = 36 + 24 = 60 > 0

√D = √60 = √4 ⋅ 15 = √2² ⋅ 15 = 2√15

x₁ = – 6√ – 2 √15 / 2 = 2 (-3 + √15) / 2 = -3 + √15

x₂ = – 6 + 2√15 / 2 = 2 (-3 + √15) / 2 = -3 + √15

x = -1; x = -3 +-√15

б) х³ + 4х² – 5 = 0

x³ + 4x² – 5 = x³ – x² + 5x² – 5 = x² (x – 1) + 5 (x² – 1) = x² ( x – 1) + 5(x – 1) ⋅ ( x + 1) = (x – 1) (x² + 5 (x + 1)) = (x – 1) ⋅ ( x² + 5x + 5)

(x – 1) ⋅ (x² + 5x + 5) = 0

x – 1 или x² + 5x + 5 = 0

x = 1

D = 5² – 4 ⋅ 1 ⋅ 5 = 25 – 20 = 5 > 0

√D = √5

x₁ = -5 – √5 / 2 = -5 – √5 / 2

x₂ = -5 + √5 / 2 = -5 + √2 / 2

x = 1; x = -5 + √5 / 2

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