Сократите дробь: а) (2m² – 8) / (m² + 6m + 8); б) (2m² – 5m + 2) / (mn – 2n – 3m + 6).

Решение:

а) (2m² – 8) / (m² + 6m + 8)

m² + 6m + 8 = 0

D = 6² ⋅ 4 ⋅ 1 ⋅ 8 = 4

m₁ = -6 + 2 / 2 = -2

m₂ = – 6 – 2 / 2 = -4

m² + 6m + 8 = (m + 2) (m + 4)

(2m² – 8) / (m² + 6m + 8) = 2 (m² – 4) / (m + 2) (m + 4) = 2 (m – 2) (m + 2) / (m + 2) (m + 4) = 2 (m – 2) / m + 4

б) (2m² – 5m + 2) / (mn – 2n – 3m + 6)

2m² – 5m + 2 = 0

D = (-5)² – 4 ⋅ 2 ⋅ 2 = 9

m₁ = 5 + 3 / 4 = 2

m₂ = 5 – 3 / 4 = 1/2

2m² – 5m + 2 = 2 (m – 2) (m – 1/2) = (m – 2) (2m – 1)

(2m² – 5m + 2) / (mn – 2n – 3m + 6) = (m – 2) ( 2m – 1) / n(m – 2) – 3 (m – 2) = (m – 2) ( 2m – 1) / (m – 2) (n -3) = 2m – 1 / n – 3

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