Сократите дробь: а) (12 – 5x – 2x²) / (15 – 10x); б) (3x² – 36x – 192) / (x² – 256).

Решение:

а) (12 – 5x – 2x²) / (15 – 10x)

12 – 5x – 2x² = -2x² – 5x + 12

D = (-5)² – 4 ⋅ (-2) ⋅ 12 = 25 + 96 = 121 > 0

√D = √121 = √11² = 11

x₁ = -(-5) – 11 / 2 ⋅ (-2) = 5 – 11 / -4 = -6 / 4 = 2/3

x₂ = -(-5) + 11 / 2 ⋅ (-2) = 5 + 11 / -4 = -16 / 4 = -4

-2x² – 5x + 12 = -2 (x – 3/2) (x – (-4)) = (-2x + 2 ⋅ 3/2) ( x + 4) = (3 – 2x) ( x + 4)

12 = 5x – 2x³ / 15 – 10x = (3 – 2x) (x + 4) / 5 (3 – 2x) = x + 4 / 5

б) (3x² – 36x – 192) / (x² – 256)

3x² – 36x – 192 = 3(x² – 12x – 64)

D = (-12)² -4 ⋅ 1 ⋅ (-64) = 144 + 256 = 400 > 0

√D = √400 = √20² = 20

x₁ = -(-12) – 20 / 20 = 12 – 20 / 2 = – 8/2 = – 4

x₂ = -(-12) + 20 / 20 = 12 + 20 / 2 = 32/2 = 16

3(x² – 12x – 64) = 3 (x – (-4)) (x – 16) = 3 (x +4) (x – 16)

3x² – 36x – 192 / x² – 256 = 3(x + 4) ( x – 16) / (x – 16) ( x + 16) = 3(x + 4) / x + 16 = 3x + 12 / x + 16

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