(x-1)²/8+8/(x-1)²=7*((x-1)/4-2/(x-1))-1 ОДЗ: x-1≠0 x≠1
(x-1)²/(4*2)+(4*2)/(x-1)²=7*((x-1)/(2*2)-2/(x-1))-1
(x-1)²/(2²*2)+(2²*2)/(x-1)²=7*((x-1)(2*2)-2/(x-1))-1
((x-1)/2)²/2+2/((x-1)/2)²=7*((x-1)/2)/2-2/(x-1))-1
Пусть (x-1)/2=t ⇒
(t²/2)+2/t²=7*(t/2-1/t)-1
(t/2-1/t)²=t²/4-2*(t/2)*(1/t)+(1/t)²=t²/4-1+(1/t)² ⇒
(t/2-1/t)²=(t²/4)-1+(1/t²)
(t/2-1/t)²+1=(t²/4)+(1/t²) |×2
2*(t/2-1/t)²+2=(t²/2)+2/t² ⇒
2*(t/2-1/t)²+2=7*(t/2-1/t)-1
Пусть t/2-1/t=v ⇒
2v²+2=7v-1
2v²-7v+3=0 D=25 √D=5
v₁=3 v₂=0,5 ⇒
1)
t/2-1/t=3 |×2t
t²-2=6t
t²-6t-2=0 D=44 √D=√44=2√22
t=3+√22 ⇒ (x-1)/2=3+√22 |×2 x-1=6+2*√22 x₁=7+2*√22
t=3-√22 ⇒ (x-1)/2=3-√22 |×2 x-1=6-2*√22 x₂=7-2*√22.
2)
v₂=0,5 ⇒
t/2-1/t=0,5 |×2t
t²-2=t
t²-t-2=0 D=9 √D=3
t₁=2 ⇒ (x-1)/2=2 |×2 x-1=4 x₃=5
t₂=-1 ⇒ (x-1)/2=-1 |×2 x-1=-2 x₄=-1
Ответ: x₁=6+2*√22 x₂=6-2*√22 x₃=5 x₄=-1.