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[Euphorica]

x^2/(x^2-1) -x^2-1/x^2 = 3/2



2x-1/x-1 = 7x-1/2(x+1)



4x+1/2x-4 = x+2/x-3



3x-15/2x-2 = 2(x-1)/x+2



x^2-x+2/x-2 = 1-(6x-8)/2-x



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



2x-1/x^3+1+1/x+1 = 2/x^2-x+1

[math10.com]

1.[tex]\frac{x^2}{x^2-1}-\frac{x^2-1}{x^2}=\frac{3}{2} ; DM:x\ne \pm 1 , x\ne 0[/tex]

Полагаме [tex]\frac{x^2}{x^2-1}=t ; =>\frac{x^2-1}{x^2}=\frac{1}{t}[/tex]

[tex]t-\frac{1}{t}=\frac{3}{2}[/tex] Подвеждаме под общ знаменател и рационализираме

[tex]2t^2-3t-2=0[/tex]

[tex]D=9+16=25[/tex]

[tex]t_1=\frac{3+5}{4}=2 ; t_2=\frac{3-5}{2}=-\frac{1}{2}[/tex]

[tex]\frac{x^2}{x^2-1}=2 ; =>2x^2-2=x^2 ; =>x^2=2 ; =>x=\pm \sqrt{2}[/tex]

[tex]\frac{x^2}{x^2-1}=-\frac{1}{2} ; =>2x^2=-x^2+1 ; =>3x^2=1 ; =>x^2=\frac{1}{3} ; =>x=\pm \frac{\sqrt{3}}{3}[/tex]

2.[tex]\frac{2x-1}{x-1}=\frac{7x-1}{2(x+1)}[/tex]

[tex]\frac{2(x+1)(2x-1)}{2(x+1)(x-1)}=\frac{(7x-1)(x-1)}{2(x+1)(x-1)} ; DM: x\ne \pm 1[/tex]

[tex]2(x+1)(2x-1)=(7x-1)(x-1)[/tex]

[tex]4x^2+2x-2=7x^2-8x+1[/tex]

[tex]3x^2-10x+3=0[/tex]

[tex]D=100-36=64[/tex]

[tex]x_1=\frac{10+8}{6}=3 ; x_2=\frac{10-8}{6}=\frac{1}{3}[/tex]

3.[tex]\frac{4x+1}{2x-4}=\frac{x+2}{x-3}[/tex]

[tex]\frac{(4x+1)(x-3)}{2(x-2)(x-3)}=\frac{2(x-2)(x+2)}{2(x-2)(x-3)} ; DM: x\ne2 , x\ne \ne[/tex]

[tex](4x+1)(x-3)=2(x-2)(x+2)[/tex]

[tex]4x^2-11x-3=2x^2-8[/tex]

[tex]2x^2-11x+5=0[/tex]

[tex]D=121-40=81[/tex]

[tex]x_1=\frac{11+9}{4}=5 ; x_2=\frac{11-9}{4}=\frac{1}{2}[/tex]

[math10.com]

4.[tex]\frac{3x-15}{2x-2}=\frac{2(x-2)}{x+2}[/tex]

[tex]\frac{3(x-5)(x+2)}{2(x-1)(x+2)}=\frac{2(x-2).2(x-1)}{2(x-1)(x+2)} ; DM: x\ne 1;x\ne -2[/tex]

[tex]3(x-5)(x+2)=2(x-2).2(x-1)[/tex]

[tex]3x^2-9x-30=4x^2-12x+8[/tex]

[tex]x^2-3x+38=0[/tex]

[tex]D=9-152=-143<0 ; =>[/tex] няма реални корени

5.[tex]\frac{x^2-x+2}{x-2}=1-\frac{6x-8}{2-x}[/tex]

[tex]\frac{x^2-x+2}{x-2}=\frac{x-2}{x-2}+\frac{6x-8}{x-2} ; DM: x\ne 2[/tex]

[tex]x^2-x+2=x-2+6x-8[/tex]

[tex]x^2-8x+12=0[/tex]

[tex]D=64-48=16[/tex]

[tex]x_1=\frac{8+4}{2}=6 ; x_2=\frac{8-4}{2}=2\notin DM[/tex]

6.[tex]\frac{6}{x^2-1}-\frac{2}{x-1}=\frac{x-2}{x+1}[/tex]

[tex]\frac{6}{(x-1)(x+1)}-\frac{2(x+1)}{(x-1)(x+1)}=\frac{(x-2)(x-1)}{(x+1)(x-1)} ; DM:x\ne \pm 1[/tex]

[tex]6-2(x+1)=(x-2)(x-1)[/tex]

[tex]6-2x-2=x^2-3x+2[/tex]

[tex]x^2-x-2=0[/tex]

[tex]D=1+8=9[/tex]

[tex]x_1=\frac{1+3}{2}=2 ; x_2=\frac{1-3}{2}=-1\notin DM[/tex]

7.[tex]\frac{2x-1}{x^3+1}+\frac{1}{x+1}=\frac{2}{x^2-x+1}[/tex]

[tex]\frac{2x-1}{(x+1)(x^2-x+1)}+\frac{x^2-x+1}{(x+1)(x^2-x+1)}=\frac{2(x+1)}{(x+1)(x^2-x+1)} ;DMx\ne -1[/tex]

[tex]2x-1+x^2-x+1=2(x+1)[/tex]

[tex]x^2-x-2=0[/tex]

[tex]D=1+8=9[/tex]

[tex]x_1=\frac{1+3}{2}=2 ; x_2=\frac{1-3}{2}=-1\notin DM[/tex]