Разложите на множители квадратный трехчлен: а) Зх² – 25х – 28; б) 2х² + 13х – 7.

Решение:

а) Зх² – 25х – 28

D = (-25)² – 4 ⋅ 3 ⋅ (-28) = 625 + 336 = 961 больше 0

√D = √961 = √31² = 31

x₁ = -(-25) – 31 / 2 ⋅ 3 = 25 – 31 / 6 = -6/6 = -1

x₂ = -(-25) + 31 / 2 ⋅ 3 = 25 + 31 / 6 = 56/6 = 28/3

3x² – 25x – 28 = 3(x – (-1)) ⋅ (x – 28/3) = (x + 1) ⋅ ( 3 ⋅ x – 3 ⋅ 28/3) = (x + 1) ⋅ (3 ⋅ x – 3 ⋅ 28/3) = (x + 1) (3x – 28)

б) 2х² + 13х – 7

D = 13² – 4 ⋅ 2 ⋅ (-7) = 169 + 56 = 225 больше 0

√D = √225 = √15² = 15

x₁ = -13 – 15 / 2 ⋅ 2 = -28 / 4 = -7

x₂ = -13 + 15 / 2 ⋅ 2 = 2 / 4 = 1/2

2x² + 13x – 7 = 2(x -(-7)) ⋅ (x – 1/2) = (x + 7) ⋅ (2 ⋅ x – 2 ⋅ 1/2) ⋅ (x + 7) ⋅ (2x – 1)

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