Решите уравнение: а) (8х – 1) ⋅ (2х – 3) – (4х – 1)² = 38; б) ((15х – 1) ⋅ (1 + 15х)) / 3 = 2 2/3; в) 0,5у³ – 0,5у ⋅ (у + 1) ⋅ (у – 3) = 7; г) x⁴ – x² = ((1 + 2х²) ⋅ (2х² -1)) / 4.

Решение:

а) (8х – 1) ⋅ (2х – 3) – (4х – 1)² = 38

8x ⋅ 2x + 8x ⋅ (-3) – 1 ⋅ 2x – 1 ⋅ (-3) – ((4x)² – 2 ⋅ 4x ⋅ 1 + 1² ) = 38

16x² – 24x – 2x + 3 – (16x² – 8x + 1) = 38

16x² – 24x – 2x + 3 – 16x² + 8x – 1 = 38

-18x = 36

x = 36 : (-18)

x = -2

б) ((15х – 1) ⋅ (1 + 15х)) / 3 = 2 2/3

(15x – 1)² / 3 = 8/3

225x² – 1 = 8

225x² = 8 + 1

225x² = 9

x² = 9/225

x = +-√9/225

x = +-3/15

x = 0,2

в) 0,5у³ – 0,5у ⋅ (у + 1) ⋅ (у – 3) = 7

0,5y³ – 0,5y(y ⋅ y + y ⋅ (-3) + 1 ⋅ y + 1 ⋅ (-3)) = 7

0,5y³ – 0,5y(y² – 3y + y – 3) = 7

0,5y³ – 0,5y (y² -2y – 3) = 7

0,5y³ -0,5y ⋅ y² – 0,5y ⋅ (-2y) – 0,5y ⋅ (-3) = 7

0,5y³ – 0,5y³ + y² + 1,5y – 7 = 0

y² + 1,5y³ – 7 = 0

2y² + 3y – 14 = 0

D = 3² – 4 ⋅ 2 ⋅ (-14) = 9 + 112 = 121 больше 0

√D = √121 = √11² = 11

y₁ = -3 – 11 / 2 ⋅ 2 = -(3 + 11)/4 = – 14/4 = – 3,5

y₂ = -3 + 11 / 2 ⋅ 2 = 11 – 3/4 = 8/4 = 2

г) x⁴ – x² = ((1 + 2х²) ⋅ (2х² -1)) / 4

x⁴ – x² = (2x²)² – 1² / 4

4x⁴ – 4x² = 4x⁴ – 1

4x⁴ – 4x² – 4x⁴ = – 1

-4x² = -1

x² = -1 :(-4)

x² = 0,25

x = +- 0,5

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