Решите уравнение х(х² + 2х + 1) = 6(х + 1).

Источник: ЕГЭ ОГЭ по математике Под редакцией И.В. Ященко.

Решение:

x(x² + 2x + 1) = 6(x + 1)

x² + 2x + 1 = 0

D = b² – 4ac

D = 2² – 4¹ ∙ 1 = 4 – 4 = 0

x = – b / (2a)

x₁ = x₂ = – 2 / 2 = -1

x² + 2x + 1 = (x + 1) ∙ (x + 1) = (x + 1)²

x ∙ (x + 1)² = 6 ∙ (x + 1)

x ∙ (x + 1)² – 6 ∙ (x + 1) = 0

(x + 1) ∙ (x(x + 1) – 6) = 0

(x + 1) ∙ (x² + x – 6) = 0

1) x + 1 = 0

x₁ = -1

2) x² + x – 6 = 0

D = 1² – 4 ∙ 1 ∙ (-6) = 1 + 24 = 25

√D = 5

x = (- b ± √D) / (2a)

x₂ = (-1 + 5) / 2 = 4 / 2 = 2

x₃ = (-1 – 5) / 2 = – 6 / 2 = – 3

Ответ: -3; -1; 2