Решите неравенство: а) х² + 2х – 48 < 0; б) 2х² – 7х + 6 > 0; в) -х² + 2х + 15 < 0; г) -5х² + 11х – 6 > 0; д) 4х² – 12х + 9 > 0; е) 25х² + З0х + 9 < 0; ж) -10х² + 9х > 0; з) -2х² + 7х < 0.

Решение:

а) х² + 2х – 48 < 0

D = 2² – 4 ⋅ 1 ⋅ (-48) = 4 + 192 = 196 > 0

√D = √196 = √14² = 14

x₁ = -2 – 14 / 2 = -(2 + 14) / 2 = – 16 / 2 = -8

x₂ = -2 + 14 / 2 = 14 – 2 / 2 = 12 / 2 = 6

б) 2х² – 7х + 6 > 0

D = (-7)² – 4 ⋅ 2 ⋅ 6 = 49 – 48 = 1 > 0

√D = √1 = 1

x₁ = -(-7) – 1 / 2 ⋅ 2 = 7 – 1 / 4 = 6 / 4 = 1,5

x₂ = -(-7) + 1 / 2 ⋅ 2 = 7 + 1 / 4 = 8 / 4 = 2

в) -х² + 2х + 15 < 0

D = 2² – 4 ⋅ (-1) ⋅ 15 = 4 + 60 = 64 > 0

√D = √64 = √8² = 8

x₁ = -2 – 8 / 2 ⋅ (-1) = -(2 + 8) / -2 = 10 / 2 = 5

x₂ = -2 + 8 / 2 ⋅ (-1) = 8 – 2 / -2 = – 6 / 2 = -3

г) -5х² + 11х – 6 > 0

D = 11² – 4 ⋅ (-5) ⋅ (-6) = 121 – 120 = 1 > 0

√D = √1 = 1

x₁ = -11 – 1 / 2 ⋅ (-5) = -(11 + 1) / – 10 = 12 / 10 = 1,2

x₂ = -11 + 1 / 2 ⋅ (-5) = -(11 – 1) / – 10 = 10 / 10 = 1

д) 4х² – 12х + 9 > 0

D = (-12)² – 4 ⋅ 4 ⋅ 9 = 144 – 144 = 0

x = -(-12) / 2 ⋅ 4 = 12 / 8 = 3/2 = 1,5

x (- ∞; 1,5) U ( 1,5; +∞)

е) 25х² + З0х + 9 < 0

D = 30² – 4 ⋅ 25 ⋅ 9 = 900 – 900 = 0

x = -30 / 2 ⋅ 25 = – 30 / 50 = 3/5 = -0,6

нет решений

ж) -10х² + 9х > 0

-10х² + 9х = 0

x (-10x + 9) = 0

x = 0 или -10x + 9 = 0

-10x = -9

x = -9: (-10)

x = 0,9

x = (0; 0,9)

з) -2х² + 7х < 0

-2х² + 7х = 0

x (-2x + 7) = 0

x = 0 или -2x + 7 = 0

-2x = -7

x = -7: (-2)

x = 3,5

x = (-∞; 0) U (3,5; +∞)

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