Atrisiniet {left(-2sqrt{1/4-x^2}x+2right)}^2+z^2={left(4-z+2sqrt{1/4-x^2}x-2right)}^2

$${ \left(-2 \sqrt{ \frac{ 1 }{ 4- { x }^{ 2 } } } x+2 \right) }^{ 2 } + { z }^{ 2 } = { \left(4-z+2 \sqrt{ \frac{ 1 }{ 4- { x }^{ 2 } } } x-2 \right) }^{ 2 }$$

Atrast z (complex solution)

$$z=\frac{4x}{\sqrt{4-x^{2}}+x}$$

$$x\neq -\sqrt{2}\text{ and }x\neq 2\text{ and }x\neq -2$$

Lineāra vienādojuma risināšanas darbības

Lietojiet Ņūtona binomu \left(a+b\right)^{2}=a^{2}+2ab+b^{2}, lai izvērstu \left(-2\sqrt{\frac{1}{4-x^{2}}}x+2\right)^{2}.

$$4\left(\sqrt{\frac{1}{4-x^{2}}}\right)^{2}x^{2}-8\sqrt{\frac{1}{4-x^{2}}}x+4+z^{2}=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Aprēķiniet \sqrt{\frac{1}{4-x^{2}}} pakāpē 2 un iegūstiet \frac{1}{4-x^{2}}.

$$4\times \frac{1}{4-x^{2}}x^{2}-8\sqrt{\frac{1}{4-x^{2}}}x+4+z^{2}=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Izsakiet 4\times \frac{1}{4-x^{2}} kā vienu daļskaitli.

$$\frac{4}{4-x^{2}}x^{2}-8\sqrt{\frac{1}{4-x^{2}}}x+4+z^{2}=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Izsakiet \frac{4}{4-x^{2}}x^{2} kā vienu daļskaitli.

$$\frac{4x^{2}}{4-x^{2}}-8\sqrt{\frac{1}{4-x^{2}}}x+4+z^{2}=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Sadaliet reizinātājos 4-x^{2}.

$$\frac{4x^{2}}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x+4+z^{2}=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Lai saskaitītu vai atņemtu izteiksmes, izvērsiet tās, vienādojot saucējus. Reiziniet 4 reiz \frac{\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)}.

$$\frac{4x^{2}}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x+\frac{4\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)}+z^{2}=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Tā kā \frac{4x^{2}}{\left(x-2\right)\left(-x-2\right)} un \frac{4\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)} ir viens un tas pats saucējs, saskaitiet tos, saskaitot to skaitītājus.

$$\frac{4x^{2}+4\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x+z^{2}=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Veiciet reizināšanas darbības izteiksmē 4x^{2}+4\left(x-2\right)\left(-x-2\right).

$$\frac{4x^{2}-4x^{2}-8x+8x+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x+z^{2}=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Apvienojiet līdzīgos locekļus izteiksmē 4x^{2}-4x^{2}-8x+8x+16.

$$\frac{16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x+z^{2}=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Lai saskaitītu vai atņemtu izteiksmes, izvērsiet tās, vienādojot saucējus. Reiziniet z^{2} reiz \frac{\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)}.

$$\frac{16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x+\frac{z^{2}\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)}=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Tā kā \frac{16}{\left(x-2\right)\left(-x-2\right)} un \frac{z^{2}\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)} ir viens un tas pats saucējs, saskaitiet tos, saskaitot to skaitītājus.

$$\frac{16+z^{2}\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Veiciet reizināšanas darbības izteiksmē 16+z^{2}\left(x-2\right)\left(-x-2\right).

$$\frac{16-z^{2}x^{2}-2z^{2}x+2z^{2}x+4z^{2}}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Apvienojiet līdzīgos locekļus izteiksmē 16-z^{2}x^{2}-2z^{2}x+2z^{2}x+4z^{2}.

$$\frac{4z^{2}-z^{2}x^{2}+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Atņemiet 2 no 4, lai iegūtu 2.

$$\frac{4z^{2}-z^{2}x^{2}+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=\left(2-z+2\sqrt{\frac{1}{4-x^{2}}}x\right)^{2}$$

Kāpiniet 2-z+2\sqrt{\frac{1}{4-x^{2}}}x kvadrātā.

$$\frac{4z^{2}-z^{2}x^{2}+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=z^{2}-4\sqrt{\frac{1}{-x^{2}+4}}xz-4z+4\left(\sqrt{\frac{1}{-x^{2}+4}}\right)^{2}x^{2}+8\sqrt{\frac{1}{-x^{2}+4}}x+4$$

Aprēķiniet \sqrt{\frac{1}{-x^{2}+4}} pakāpē 2 un iegūstiet \frac{1}{-x^{2}+4}.

$$\frac{4z^{2}-z^{2}x^{2}+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=z^{2}-4\sqrt{\frac{1}{-x^{2}+4}}xz-4z+4\times \frac{1}{-x^{2}+4}x^{2}+8\sqrt{\frac{1}{-x^{2}+4}}x+4$$

Izsakiet 4\times \frac{1}{-x^{2}+4} kā vienu daļskaitli.

$$\frac{4z^{2}-z^{2}x^{2}+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=z^{2}-4\sqrt{\frac{1}{-x^{2}+4}}xz-4z+\frac{4}{-x^{2}+4}x^{2}+8\sqrt{\frac{1}{-x^{2}+4}}x+4$$

Izsakiet \frac{4}{-x^{2}+4}x^{2} kā vienu daļskaitli.

$$\frac{4z^{2}-z^{2}x^{2}+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=z^{2}-4\sqrt{\frac{1}{-x^{2}+4}}xz-4z+\frac{4x^{2}}{-x^{2}+4}+8\sqrt{\frac{1}{-x^{2}+4}}x+4$$

Sadaliet reizinātājos -x^{2}+4.

$$\frac{4z^{2}-z^{2}x^{2}+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=z^{2}-4z+4-4\sqrt{\frac{1}{-x^{2}+4}}xz+\frac{4x^{2}}{\left(x-2\right)\left(-x-2\right)}+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Lai saskaitītu vai atņemtu izteiksmes, izvērsiet tās, vienādojot saucējus. Reiziniet z^{2}-4z+4 reiz \frac{\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)}.

$$\frac{4z^{2}-z^{2}x^{2}+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=\frac{\left(z^{2}-4z+4\right)\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)}-4\sqrt{\frac{1}{-x^{2}+4}}xz+\frac{4x^{2}}{\left(x-2\right)\left(-x-2\right)}+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Tā kā \frac{\left(z^{2}-4z+4\right)\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)} un \frac{4x^{2}}{\left(x-2\right)\left(-x-2\right)} ir viens un tas pats saucējs, saskaitiet tos, saskaitot to skaitītājus.

$$\frac{4z^{2}-z^{2}x^{2}+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=\frac{\left(z^{2}-4z+4\right)\left(x-2\right)\left(-x-2\right)+4x^{2}}{\left(x-2\right)\left(-x-2\right)}-4\sqrt{\frac{1}{-x^{2}+4}}xz+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Veiciet reizināšanas darbības izteiksmē \left(z^{2}-4z+4\right)\left(x-2\right)\left(-x-2\right)+4x^{2}.

$$\frac{4z^{2}-z^{2}x^{2}+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=\frac{4z^{2}-z^{2}x^{2}-16z+4zx^{2}-4x^{2}+16+4x^{2}}{\left(x-2\right)\left(-x-2\right)}-4\sqrt{\frac{1}{-x^{2}+4}}xz+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Apvienojiet līdzīgos locekļus izteiksmē 4z^{2}-z^{2}x^{2}-16z+4zx^{2}-4x^{2}+16+4x^{2}.

$$\frac{4z^{2}-z^{2}x^{2}+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=\frac{16-z^{2}x^{2}+4z^{2}+4zx^{2}-16z}{\left(x-2\right)\left(-x-2\right)}-4\sqrt{\frac{1}{-x^{2}+4}}xz+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Izmantojiet distributīvo īpašību, lai reizinātu x-2 ar -x-2 un apvienotu līdzīgos locekļus.

$$\frac{4z^{2}-z^{2}x^{2}+16}{-x^{2}+4}-8\sqrt{\frac{1}{4-x^{2}}}x=\frac{16-z^{2}x^{2}+4z^{2}+4zx^{2}-16z}{\left(x-2\right)\left(-x-2\right)}-4\sqrt{\frac{1}{-x^{2}+4}}xz+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Izmantojiet distributīvo īpašību, lai reizinātu x-2 ar -x-2 un apvienotu līdzīgos locekļus.

$$\frac{4z^{2}-z^{2}x^{2}+16}{-x^{2}+4}-8\sqrt{\frac{1}{4-x^{2}}}x=\frac{16-z^{2}x^{2}+4z^{2}+4zx^{2}-16z}{-x^{2}+4}-4\sqrt{\frac{1}{-x^{2}+4}}xz+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Atņemiet \frac{16-z^{2}x^{2}+4z^{2}+4zx^{2}-16z}{-x^{2}+4} no abām pusēm.

$$\frac{4z^{2}-z^{2}x^{2}+16}{-x^{2}+4}-8\sqrt{\frac{1}{4-x^{2}}}x-\frac{16-z^{2}x^{2}+4z^{2}+4zx^{2}-16z}{-x^{2}+4}=-4\sqrt{\frac{1}{-x^{2}+4}}xz+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Tā kā \frac{4z^{2}-z^{2}x^{2}+16}{-x^{2}+4} un \frac{16-z^{2}x^{2}+4z^{2}+4zx^{2}-16z}{-x^{2}+4} ir viens un tas pats saucējs, atņemiet tos, atņemot to skaitītājus.

$$\frac{4z^{2}-z^{2}x^{2}+16-\left(16-z^{2}x^{2}+4z^{2}+4zx^{2}-16z\right)}{-x^{2}+4}-8\sqrt{\frac{1}{4-x^{2}}}x=-4\sqrt{\frac{1}{-x^{2}+4}}xz+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Veiciet reizināšanas darbības izteiksmē 4z^{2}-z^{2}x^{2}+16-\left(16-z^{2}x^{2}+4z^{2}+4zx^{2}-16z\right).

$$\frac{4z^{2}-z^{2}x^{2}+16-16+z^{2}x^{2}-4z^{2}-4zx^{2}+16z}{-x^{2}+4}-8\sqrt{\frac{1}{4-x^{2}}}x=-4\sqrt{\frac{1}{-x^{2}+4}}xz+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Apvienojiet līdzīgos locekļus izteiksmē 4z^{2}-z^{2}x^{2}+16-16+z^{2}x^{2}-4z^{2}-4zx^{2}+16z.

$$\frac{16z-4zx^{2}}{-x^{2}+4}-8\sqrt{\frac{1}{4-x^{2}}}x=-4\sqrt{\frac{1}{-x^{2}+4}}xz+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Sadaliet reizinātājos izteiksmes, kas vēl nav sadalītas reizinātājos formulā \frac{16z-4zx^{2}}{-x^{2}+4}.

$$\frac{4z\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=-4\sqrt{\frac{1}{-x^{2}+4}}xz+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Saīsiniet \left(x-2\right)\left(-x-2\right) gan skaitītājā, gan saucējā.

$$4z-8\sqrt{\frac{1}{4-x^{2}}}x=-4\sqrt{\frac{1}{-x^{2}+4}}xz+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Pievienot 4\sqrt{\frac{1}{-x^{2}+4}}xz abās pusēs.

$$4z-8\sqrt{\frac{1}{4-x^{2}}}x+4\sqrt{\frac{1}{-x^{2}+4}}xz=8\sqrt{\frac{1}{-x^{2}+4}}x$$

Pievienot 8\sqrt{\frac{1}{4-x^{2}}}x abās pusēs.

$$4z+4\sqrt{\frac{1}{-x^{2}+4}}xz=8\sqrt{\frac{1}{-x^{2}+4}}x+8\sqrt{\frac{1}{4-x^{2}}}x$$

Savelciet 8\sqrt{\frac{1}{-x^{2}+4}}x un 8\sqrt{\frac{1}{4-x^{2}}}x, lai iegūtu 16\sqrt{\frac{1}{-x^{2}+4}}x.

$$4z+4\sqrt{\frac{1}{-x^{2}+4}}xz=16\sqrt{\frac{1}{-x^{2}+4}}x$$

Savelciet visus locekļus, kuros ir z.

$$\left(4+4\sqrt{\frac{1}{-x^{2}+4}}x\right)z=16\sqrt{\frac{1}{-x^{2}+4}}x$$

Vienādojums ir standarta formā.

$$\left(4\sqrt{\frac{1}{4-x^{2}}}x+4\right)z=16\sqrt{\frac{1}{4-x^{2}}}x$$

Daliet abas puses ar 4+4\sqrt{\left(-x^{2}+4\right)^{-1}}x.

$$\frac{\left(4\sqrt{\frac{1}{4-x^{2}}}x+4\right)z}{4\sqrt{\frac{1}{4-x^{2}}}x+4}=\frac{16\left(4-x^{2}\right)^{-\frac{1}{2}}x}{4\sqrt{\frac{1}{4-x^{2}}}x+4}$$

Dalīšana ar 4+4\sqrt{\left(-x^{2}+4\right)^{-1}}x atsauc reizināšanu ar 4+4\sqrt{\left(-x^{2}+4\right)^{-1}}x.

$$z=\frac{16\left(4-x^{2}\right)^{-\frac{1}{2}}x}{4\sqrt{\frac{1}{4-x^{2}}}x+4}$$

Daliet 16\left(-x^{2}+4\right)^{-\frac{1}{2}}x ar 4+4\sqrt{\left(-x^{2}+4\right)^{-1}}x.

$$z=\frac{4x}{\sqrt{4-x^{2}}+x}$$

Atrast z

$$z=\frac{4x}{\sqrt{4-x^{2}}+x}$$

$$x\neq -\sqrt{2}\text{ and }|x|<2$$

Lineāra vienādojuma risināšanas darbības

Lietojiet Ņūtona binomu \left(a+b\right)^{2}=a^{2}+2ab+b^{2}, lai izvērstu \left(-2\sqrt{\frac{1}{4-x^{2}}}x+2\right)^{2}.

$$4\left(\sqrt{\frac{1}{4-x^{2}}}\right)^{2}x^{2}-8\sqrt{\frac{1}{4-x^{2}}}x+4+z^{2}=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Aprēķiniet \sqrt{\frac{1}{4-x^{2}}} pakāpē 2 un iegūstiet \frac{1}{4-x^{2}}.

$$4\times \frac{1}{4-x^{2}}x^{2}-8\sqrt{\frac{1}{4-x^{2}}}x+4+z^{2}=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Izsakiet 4\times \frac{1}{4-x^{2}} kā vienu daļskaitli.

$$\frac{4}{4-x^{2}}x^{2}-8\sqrt{\frac{1}{4-x^{2}}}x+4+z^{2}=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Izsakiet \frac{4}{4-x^{2}}x^{2} kā vienu daļskaitli.

$$\frac{4x^{2}}{4-x^{2}}-8\sqrt{\frac{1}{4-x^{2}}}x+4+z^{2}=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Sadaliet reizinātājos 4-x^{2}.

$$\frac{4x^{2}}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x+4+z^{2}=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Lai saskaitītu vai atņemtu izteiksmes, izvērsiet tās, vienādojot saucējus. Reiziniet 4 reiz \frac{\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)}.

$$\frac{4x^{2}}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x+\frac{4\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)}+z^{2}=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Tā kā \frac{4x^{2}}{\left(x-2\right)\left(-x-2\right)} un \frac{4\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)} ir viens un tas pats saucējs, saskaitiet tos, saskaitot to skaitītājus.

$$\frac{4x^{2}+4\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x+z^{2}=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Veiciet reizināšanas darbības izteiksmē 4x^{2}+4\left(x-2\right)\left(-x-2\right).

$$\frac{4x^{2}-4x^{2}-8x+8x+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x+z^{2}=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Apvienojiet līdzīgos locekļus izteiksmē 4x^{2}-4x^{2}-8x+8x+16.

$$\frac{16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x+z^{2}=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Lai saskaitītu vai atņemtu izteiksmes, izvērsiet tās, vienādojot saucējus. Reiziniet z^{2} reiz \frac{\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)}.

$$\frac{16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x+\frac{z^{2}\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)}=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Tā kā \frac{16}{\left(x-2\right)\left(-x-2\right)} un \frac{z^{2}\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)} ir viens un tas pats saucējs, saskaitiet tos, saskaitot to skaitītājus.

$$\frac{16+z^{2}\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Veiciet reizināšanas darbības izteiksmē 16+z^{2}\left(x-2\right)\left(-x-2\right).

$$\frac{16-z^{2}x^{2}-2z^{2}x+2z^{2}x+4z^{2}}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Apvienojiet līdzīgos locekļus izteiksmē 16-z^{2}x^{2}-2z^{2}x+2z^{2}x+4z^{2}.

$$\frac{4z^{2}-z^{2}x^{2}+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=\left(4-z+2\sqrt{\frac{1}{4-x^{2}}}x-2\right)^{2}$$

Atņemiet 2 no 4, lai iegūtu 2.

$$\frac{4z^{2}-z^{2}x^{2}+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=\left(2-z+2\sqrt{\frac{1}{4-x^{2}}}x\right)^{2}$$

Kāpiniet 2-z+2\sqrt{\frac{1}{4-x^{2}}}x kvadrātā.

$$\frac{4z^{2}-z^{2}x^{2}+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=z^{2}-4\sqrt{\frac{1}{-x^{2}+4}}xz-4z+4\left(\sqrt{\frac{1}{-x^{2}+4}}\right)^{2}x^{2}+8\sqrt{\frac{1}{-x^{2}+4}}x+4$$

Aprēķiniet \sqrt{\frac{1}{-x^{2}+4}} pakāpē 2 un iegūstiet \frac{1}{-x^{2}+4}.

$$\frac{4z^{2}-z^{2}x^{2}+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=z^{2}-4\sqrt{\frac{1}{-x^{2}+4}}xz-4z+4\times \frac{1}{-x^{2}+4}x^{2}+8\sqrt{\frac{1}{-x^{2}+4}}x+4$$

Izsakiet 4\times \frac{1}{-x^{2}+4} kā vienu daļskaitli.

$$\frac{4z^{2}-z^{2}x^{2}+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=z^{2}-4\sqrt{\frac{1}{-x^{2}+4}}xz-4z+\frac{4}{-x^{2}+4}x^{2}+8\sqrt{\frac{1}{-x^{2}+4}}x+4$$

Izsakiet \frac{4}{-x^{2}+4}x^{2} kā vienu daļskaitli.

$$\frac{4z^{2}-z^{2}x^{2}+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=z^{2}-4\sqrt{\frac{1}{-x^{2}+4}}xz-4z+\frac{4x^{2}}{-x^{2}+4}+8\sqrt{\frac{1}{-x^{2}+4}}x+4$$

Sadaliet reizinātājos -x^{2}+4.

$$\frac{4z^{2}-z^{2}x^{2}+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=z^{2}-4z+4-4\sqrt{\frac{1}{-x^{2}+4}}xz+\frac{4x^{2}}{\left(x-2\right)\left(-x-2\right)}+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Lai saskaitītu vai atņemtu izteiksmes, izvērsiet tās, vienādojot saucējus. Reiziniet z^{2}-4z+4 reiz \frac{\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)}.

$$\frac{4z^{2}-z^{2}x^{2}+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=\frac{\left(z^{2}-4z+4\right)\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)}-4\sqrt{\frac{1}{-x^{2}+4}}xz+\frac{4x^{2}}{\left(x-2\right)\left(-x-2\right)}+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Tā kā \frac{\left(z^{2}-4z+4\right)\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)} un \frac{4x^{2}}{\left(x-2\right)\left(-x-2\right)} ir viens un tas pats saucējs, saskaitiet tos, saskaitot to skaitītājus.

$$\frac{4z^{2}-z^{2}x^{2}+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=\frac{\left(z^{2}-4z+4\right)\left(x-2\right)\left(-x-2\right)+4x^{2}}{\left(x-2\right)\left(-x-2\right)}-4\sqrt{\frac{1}{-x^{2}+4}}xz+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Veiciet reizināšanas darbības izteiksmē \left(z^{2}-4z+4\right)\left(x-2\right)\left(-x-2\right)+4x^{2}.

$$\frac{4z^{2}-z^{2}x^{2}+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=\frac{4z^{2}-z^{2}x^{2}-16z+4zx^{2}-4x^{2}+16+4x^{2}}{\left(x-2\right)\left(-x-2\right)}-4\sqrt{\frac{1}{-x^{2}+4}}xz+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Apvienojiet līdzīgos locekļus izteiksmē 4z^{2}-z^{2}x^{2}-16z+4zx^{2}-4x^{2}+16+4x^{2}.

$$\frac{4z^{2}-z^{2}x^{2}+16}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=\frac{16-z^{2}x^{2}+4z^{2}+4zx^{2}-16z}{\left(x-2\right)\left(-x-2\right)}-4\sqrt{\frac{1}{-x^{2}+4}}xz+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Izmantojiet distributīvo īpašību, lai reizinātu x-2 ar -x-2 un apvienotu līdzīgos locekļus.

$$\frac{4z^{2}-z^{2}x^{2}+16}{-x^{2}+4}-8\sqrt{\frac{1}{4-x^{2}}}x=\frac{16-z^{2}x^{2}+4z^{2}+4zx^{2}-16z}{\left(x-2\right)\left(-x-2\right)}-4\sqrt{\frac{1}{-x^{2}+4}}xz+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Izmantojiet distributīvo īpašību, lai reizinātu x-2 ar -x-2 un apvienotu līdzīgos locekļus.

$$\frac{4z^{2}-z^{2}x^{2}+16}{-x^{2}+4}-8\sqrt{\frac{1}{4-x^{2}}}x=\frac{16-z^{2}x^{2}+4z^{2}+4zx^{2}-16z}{-x^{2}+4}-4\sqrt{\frac{1}{-x^{2}+4}}xz+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Atņemiet \frac{16-z^{2}x^{2}+4z^{2}+4zx^{2}-16z}{-x^{2}+4} no abām pusēm.

$$\frac{4z^{2}-z^{2}x^{2}+16}{-x^{2}+4}-8\sqrt{\frac{1}{4-x^{2}}}x-\frac{16-z^{2}x^{2}+4z^{2}+4zx^{2}-16z}{-x^{2}+4}=-4\sqrt{\frac{1}{-x^{2}+4}}xz+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Tā kā \frac{4z^{2}-z^{2}x^{2}+16}{-x^{2}+4} un \frac{16-z^{2}x^{2}+4z^{2}+4zx^{2}-16z}{-x^{2}+4} ir viens un tas pats saucējs, atņemiet tos, atņemot to skaitītājus.

$$\frac{4z^{2}-z^{2}x^{2}+16-\left(16-z^{2}x^{2}+4z^{2}+4zx^{2}-16z\right)}{-x^{2}+4}-8\sqrt{\frac{1}{4-x^{2}}}x=-4\sqrt{\frac{1}{-x^{2}+4}}xz+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Veiciet reizināšanas darbības izteiksmē 4z^{2}-z^{2}x^{2}+16-\left(16-z^{2}x^{2}+4z^{2}+4zx^{2}-16z\right).

$$\frac{4z^{2}-z^{2}x^{2}+16-16+z^{2}x^{2}-4z^{2}-4zx^{2}+16z}{-x^{2}+4}-8\sqrt{\frac{1}{4-x^{2}}}x=-4\sqrt{\frac{1}{-x^{2}+4}}xz+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Apvienojiet līdzīgos locekļus izteiksmē 4z^{2}-z^{2}x^{2}+16-16+z^{2}x^{2}-4z^{2}-4zx^{2}+16z.

$$\frac{16z-4zx^{2}}{-x^{2}+4}-8\sqrt{\frac{1}{4-x^{2}}}x=-4\sqrt{\frac{1}{-x^{2}+4}}xz+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Sadaliet reizinātājos izteiksmes, kas vēl nav sadalītas reizinātājos formulā \frac{16z-4zx^{2}}{-x^{2}+4}.

$$\frac{4z\left(x-2\right)\left(-x-2\right)}{\left(x-2\right)\left(-x-2\right)}-8\sqrt{\frac{1}{4-x^{2}}}x=-4\sqrt{\frac{1}{-x^{2}+4}}xz+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Saīsiniet \left(x-2\right)\left(-x-2\right) gan skaitītājā, gan saucējā.

$$4z-8\sqrt{\frac{1}{4-x^{2}}}x=-4\sqrt{\frac{1}{-x^{2}+4}}xz+8\sqrt{\frac{1}{-x^{2}+4}}x$$

Pievienot 4\sqrt{\frac{1}{-x^{2}+4}}xz abās pusēs.

$$4z-8\sqrt{\frac{1}{4-x^{2}}}x+4\sqrt{\frac{1}{-x^{2}+4}}xz=8\sqrt{\frac{1}{-x^{2}+4}}x$$

Pievienot 8\sqrt{\frac{1}{4-x^{2}}}x abās pusēs.

$$4z+4\sqrt{\frac{1}{-x^{2}+4}}xz=8\sqrt{\frac{1}{-x^{2}+4}}x+8\sqrt{\frac{1}{4-x^{2}}}x$$

Savelciet 8\sqrt{\frac{1}{-x^{2}+4}}x un 8\sqrt{\frac{1}{4-x^{2}}}x, lai iegūtu 16\sqrt{\frac{1}{-x^{2}+4}}x.

$$4z+4\sqrt{\frac{1}{-x^{2}+4}}xz=16\sqrt{\frac{1}{-x^{2}+4}}x$$

Savelciet visus locekļus, kuros ir z.

$$\left(4+4\sqrt{\frac{1}{-x^{2}+4}}x\right)z=16\sqrt{\frac{1}{-x^{2}+4}}x$$

Vienādojums ir standarta formā.

$$\left(4\sqrt{\frac{1}{4-x^{2}}}x+4\right)z=16\sqrt{\frac{1}{4-x^{2}}}x$$

Daliet abas puses ar 4+4\sqrt{\left(-x^{2}+4\right)^{-1}}x.

$$\frac{\left(4\sqrt{\frac{1}{4-x^{2}}}x+4\right)z}{4\sqrt{\frac{1}{4-x^{2}}}x+4}=\frac{16x}{\sqrt{4-x^{2}}\left(4\sqrt{\frac{1}{4-x^{2}}}x+4\right)}$$

Dalīšana ar 4+4\sqrt{\left(-x^{2}+4\right)^{-1}}x atsauc reizināšanu ar 4+4\sqrt{\left(-x^{2}+4\right)^{-1}}x.

$$z=\frac{16x}{\sqrt{4-x^{2}}\left(4\sqrt{\frac{1}{4-x^{2}}}x+4\right)}$$

Daliet \frac{16x}{\sqrt{4-x^{2}}} ar 4+4\sqrt{\left(-x^{2}+4\right)^{-1}}x.

$$z=\frac{4x}{\sqrt{4-x^{2}}+x}$$

Atrast x (complex solution)

$$\left\{\begin{matrix}x=\sqrt{2}\left(z^{2}-4z+8\right)^{-\frac{1}{2}}z\text{, }&arg(-\left(2\left(2z^{2}-8z+16\right)^{-\frac{1}{2}}z-8\left(2z^{2}-8z+16\right)^{-\frac{1}{2}}\right))<\pi \text{ and }z\neq 4\text{ and }z\neq 2+2i\text{ and }z\neq 2-2i\\x=-\sqrt{2}\left(z^{2}-4z+8\right)^{-\frac{1}{2}}z\text{, }&z\neq 4\text{ and }z\neq 2+2i\text{ and }z\neq 2-2i\text{ and }\left(z=0\text{ or }arg(2\left(2z^{2}-8z+16\right)^{-\frac{1}{2}}z-8\left(2z^{2}-8z+16\right)^{-\frac{1}{2}})<\pi \right)\end{matrix}\right,$$

Atrast x

$$\left\{\begin{matrix}x=-\sqrt{\frac{2}{z^{2}-4z+8}}z\text{, }&z=0\text{ or }z>4\\x=\sqrt{\frac{2}{z^{2}-4z+8}}z\text{, }&z<4\end{matrix}\right,$$