Question

Rozwiąż równania:
[tex]a) 1+\frac{1}{x-1}=\frac{2}{x-2}[/tex]
[tex]b) 2-\frac{1}{x}=\frac{x}{x+2}[/tex]

Answer 1

Δ=16-8=8

√Δ=√8=2√2

x1=(4-2√2)/2=2-√2

x2=(4+2√2)/2=2+√2

Δ=9+8=17

√Δ=√17

x1=(-3-√17)/2

x2=(-3+√17)/2

Answer 2

a)

1+1/(x-1)=2/(x-2)   , dla x≠1 ∧  x≠2

(x-1+1)/(x-1)=2/(x-2)

x/(x-1)=2/(x-2)

2(x-1)=x(x-2)

2x-2=x²-2x

x²-2x-2x+2=0

x²-4x+2=0

Δ=(-4)²-4·1·2=16-8=8 , √Δ=√8=2√2

x1=(4-2√2)/2

x1=2-√2

x2=(4+2√2)/2

x2=2+√2

x∈{2-√2,2+√2}

b)

2-1/x=x/(x+2)  ,    dla x≠0  ∧    x≠-2

(2x-1)/x=x(x+2)

x²=(2x-1)(x+2)

x²=2x²+4x-x-2

x²-2x²-4x+x+2=0

-x²-3x+2=0|·(-1)

x²+3x-2=0

Δ=3²-4·1·(-2)=9+8=17 , √Δ=√17

x1=(-3-√17)/2

x2=(-3+√17)/2

x∈{(-3-√17)/2 ,(-3+√17)/2}