Question

DAUU CORONITA!!!
1) Rezolvati ecuatiile:
a) 2x-1supra x-1= 3supra2
b) x+1supra x-1= 2 1supra12
c) x+2spupra x-2 - 2-xsupra x+2=3
d) 1supra x-2 +1suprax-5 + 3x²supra4(x-2)(x-5)=0
2) Rezolvati ecuatiile:
a) 2x-1supraa x+7=3x+4supra x-1
b) x-1supra x²-1=xsupra6
c) xsupra x-2 +5supra x+2=8supra x²-4
d) x²-1supra2 =5(x-2)
e) (x-1)²supra3=2x-5
f) x²+2supra x²+4=3supra4
AJUTATI-MAA!!

Answer 1

a)( 2x-1)/(x-1)= 3/2
2(2x-1)=3(x-1)
4x-2=3x-3
4x-3x=-3+2
x=-1

b) (x+1)/(x-1)= 2 1supra12
(x+1)/(x-1)=25/12
12(x+1)=25(x-1)
12x+12=25x-25
12x-25x=-25-12
-13x=-37    inmultim -1
x=37/13

c) (x+2)/( x-2) -( 2-x)/(x+2)=3
[(x+2)(x+2)-(2-x)(x-2)]/(x-2)(x+2)=3
(x²+2x+2x+4-2x+4+x²-2x)/(x²-4)=3
2x²+8=3(x²-4)
2x²+8=3x²-12
2x²-3x²=-12-8
-x²=-20    inmultim -1
x²=20
x1=√20=2√2
x2=-2√2

d) 1/( x-2) +1/(x-5) + 3x²/4(x-2)(x-5)=0 
[x-5+x-2+3x²]/4((x-2)(-5)=0
(3x²+2x-7)/4(x-2)(x-5)=0
cond:x≠2 si x≠5
3x²+2x-7=0
Δ=4+84=88
x1=(-2+2√22)/6=2(-1+√22)/6=(-1+√22)/3
x2=(-1-√22)/3

2) Rezolvati ecuatiile:
a) (2x-1)/(x+7)=(3x+4)/( x-1)
(2x-1)((x-1)=(3x+4)(x+7)
2x²-2x-x+1=3x²+21x+4x+28
2x²-3x²-3x-25x+1-28=0
-x²-28x-27=0    inmultim -1
x²+28x+27=0
Δ=784-108=676
x1=(-28+26)/2=1
x2=(-28-26)/2=-54/2=-27

b) (x-1)/( x²-1)=x/6
6(x-1)=x(x²-1)
6x-6=x³-x
x³-x-6x-6=0
x³-7x-6=0
(x+1)(x²-x-6)=0
x1=-1
x²-x-6=0
Δ=1+24=25
x2=(1-5)/2=-4/2=-2
x2=(1+5)/2=6/2=3
S={-2,-1,3

c) (xsupra x-2 +5supra x+2=8supra x²-4
[x(x+2)+5(x-2)]/(x-2)(x+2)=8/(x²-4)
(x²+2x+5x-10)/(x²-4)=8/(x²-4)
x²+7x-10=8
x²+7x-10-8=0
x²+7x-18=0
Δ=49+72=121
x1=(-7+11)/2=4/2=2
x2=(-7-11)/2=-18/2=-9

d) x²-1supra2 =5(x-2) 
(x²-1)=10(x-2)
x²-1=10x-20
x²-10x-1+21=0
x²-10x+20=0
Δ=100-80=20
x1=(10-2√2)/2=2(5-√2)/2=5-√2
x2=5+√2

e) (x-1)²supra3=2x-5 
x²-2x+1=3(2x-5)
x²-2x+1=6x-15
x²-2x-6x+1+15=0
x²-8x+16=0
Δ=64-64=0
x1=x2=8/2=4

f) x²+2supra x²+4=3supra4 
4(x²+2)=3(x²+4)
4x²+8=3x²+12
4x²-3x²=12-8
x²=4
x1=√4=2
x2=-√4=-2