Алгебра 9 класс учебник Макарычев, Миндюк ответы – номер 242

Замена: х² – 2х = t
12/t + 4 = t − 1
ОДЗ:
t + 3 ≠ 0
t ≠ −3
12 = (t + 3)(t – 1)
12 = t² – t + 3t – 3
t² + 2t – 3 – 12 = 0
t² + 2t – 15 = 0
D = 2² – 4 · (–15) = 4 + 60 = 64

х² – 2х = 3
х² – 2х – 3 = 0
D = (–2)² – 4 · 1 · (–3) = 4 + 12 = 16

х² – 2х = –5
х² – 2х + 5 = 0
D = (–2)² – 4 · 1 · 5 = 4 – 20 = –16 – корней нет

Ответ: х = –1; х = 3

Замена: х² + х = t

12/t – 10 – 6/t – 6 = 5/t – 11 | · (t – 10)(t – 6)(t – 11)

ОДЗ:
t – 10  или  t – 6  или   t – 11
t ≠ 10           t ≠ 6           t ≠ 11

12(t – 6)(t – 11) – 6(t – 10)(t – 11) = 5(t – 10)(t – 6)

12(t² – 11t – 6t + 66) – 6(t² – 11t – 10t + 110) = 5(t² – 6t – 10t + 60)

12t² – 132t – 72t + 792 – 6t² + 66t + 60t – 660 = 5t² – 30t – 50t + 300

6t² – 78t + 132 – 5t² + 80t – 300 = 0

t² + 2t – 168 = 0

D = 2² – 4 · 1 · (–168) = 4 + 672 = 676

x² + х = 12                 х² + х = –14
х² + х – 12 = 0           х² + х + 14 = 0
По теореме Виета   D = 1² – 4 · 14 = 1 – 56 = –55 – корней нет

х₁ = 3
х₂ = –4

Ответ: х = –4; х = 3

Замена: х² – 2х = t

16/t – 11/t + 3 = 9/t + 1 | · t(t + 3)(t + 1)

ОДЗ:
t ≠ 0  или  t + 3 ≠ 0  или  t + 1 ≠ 0
                   t ≠ –3                t ≠ –1

16(t + 3)(t + 1) – 11t(t + 1) = 9t(t + 3)

16(t² + t + 3t + 3) – 11t² – 11t = 9t² + 27t

16t² + 64t + 48 – 11t² – 11t – 9t² – 27t = 0

–4t² + 26t + 48 = 0 | : (–2)

2t² – 13t – 24 = 0

D = (–13)² – 4 · 2 · (–24) = 169 + 192 = 361

х² – 2х = 8                 х² – 2х = –1,25
х² – 2х – 8 = 0           х² – 2х + 1,25 = 0
По теореме Виета   D = (–2)² · 4 · 1,25 = 4 – 5 = –1 – корней нет

х₁ = –2
х₂ = 4

Ответ: х = –2; х = 4