Решите уравнение: а) (2 / x – 2) – (10 / x + 3) = ((50 / (x² + x – 6)) – 1; б) ((x + 5) / (x – 1)) + ((2x – 5) / (x – 7)) – ((30 – 12x) / (8x – x² – 7)) = 0.

Решение:

а) (2 / x – 2) – (10 / x + 3) = ((50 / (x² + x – 6)) – 1

2(x + 3) – 10(x – 2) / x² + x – 6 = 50 – (x² + x – 6) / x² + x – 6

2x + 6 – 10x + 20 = 50 – x² – x + 6

-8x + 26 = – x² – x + 56

x² – 8x + x + 26 – 56 = 0

x² – 7x – 30 = 0

x₁ ⋅ x₂ = – 30

x₁ + x₂ = 7

x₁ = -3

x₂ = 10

x = – 3

x² + x – 6 = (-3)² + (-3) – 6 = 9 – 3 – 6 = 0

Тогда x = 10

x² + x – 6 = 10² + 10 – 6 = 100 + 4 = 104

x = 10

б) ((x + 5) / (x – 1)) + ((2x – 5) / (x – 7)) – ((30 – 12x) / (8x – x² – 7)) = 0

x + 5 / x – 1 + 2x – 5 / x – 7 – 30 – 12x / -(x² – 8x + 7) = 0

x + 5 / x – 1 + 2x – 5 / x – 7 + 30 – 12x / x² – 8x + 8 = 0

(x + 5) (x – 7) + (2x – 5) (x – 1) + 30 – 12x = 0

x² + 5x – 7x – 35 + 2x² – 2x – 5x + 5 + 30 – 12x = 0

3x² – 21x = 0

3x(x – 7) = 0

x = 0 или x – 7 = 0

x = 7

Тогда x = 0

x² – 8x + 7 = 0² – 8 ⋅ 0 + 7 = 7

Тогда x = 7

x² – 8x + 7 = 7² – 8 ⋅ 7 + 7 = 49 – 56 + 7 = 0

x = 0

Ответ: а) x = 10 ; б) x = 0

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