Question

Zad1
Rozwiąż równanie:
a)√2x-1 / √2x+1 =√2x+1
b)3/x+2 = 1/x-2
c) 1/x + x/x+2 =1
d) 1/x-1 + 1/x+2= 3/(x-1)(x+2)

Answer 1

A)√2x-1 / √2x+1 =√2x+1

(1+√2x)(1-√2x) = √2x+1
1-4x=√2x+1
-4x=√2x
(-4x) 2 = 2x
16x = 2x
14x=0
x=0


b)3/x+2 = 1/x-2 / *(x-2)
3/x+2 *x-2 =1
3x-6/x(kwadrat)-4 = 1 /*x(kwadrat)-4
3x-6=3/x(kwadrat)-4
3x=x(kwadrat)+2
x=x(kwadrat)+2/3


c) 1/x + x/x+2 =1 / (x+2)
x+2/x(kwadrat)+2+x=x+2
x+2/x(kwadrat)+2=2 /:x(kwadrat)+2
x+2=x(kwadrat)+4
x=x+(kwadrat)+2

d) 1/x-1 + 1/x+2= 3/(x-1)(x+2) /*(x+2)
1/0,5+1=3/0,5
równanie sprzeczne

Answer 2

A)√2x-1 / √2x+1 =√2x+1
b)3/x+2 = 1/x-2
c) 1/x + x/x+2 =1
d) 1/x-1 + 1/x+2= 3/(x-1)(x+2)

a)√2x-1-√2x+1=√2x+1 /*(√2x+1)
√2x-1= (√2x+1)(√2x+1)
√2x-1= 2x²+√2x+√2x+1
√2x-1= 2x²+2√2+1
√2x-2√2x-1-1-2x²=0
2x²-√2x-2=0

Δ=b²-4ac
Δ= 2+16
Δ= 18
√Δ= 3√2

x₁= -b-√Δ/2a
x₁= √2-3√2/4= √2/2

x₂= -b+√Δ/2a
x₂= √2+3√2/4=4√2/4=√2

x∈ (√2/2,√2)

b)3/x+2 = 1/x-2
3/x-1/x=-2-2
2/x=-4 /*x
2=-4x
x=-1/2

c)1/x + x/x+2 =1 /x
1+x+2x=x
1=-3x
x=-1/3

d)1/x-1 + 1/x+2= 3/(x-1)(x+2)
1(x+2)/(x-1)(x+2) + 1(x-1)/(x-1)(x+2)= 3/(x-1)(x+2)
x+2/(x-1)x+2)+ x-1/(x-1)(x+2)= 3/(x-1)(x+2)
2x-1/(x-1)(x+2) - 3/(x-1)(x+2)=0
2x-4/(x-1)(x+2)=0 /*(x-1)(x+2)
2x-4=0
2x=4
x=2