Whakaoti i te 2/x^2-1-2/x^2+1=4/x^4-1

$$\frac { 2 } { x ^ { 2 } - 1 } - \frac { 2 } { x ^ { 2 } + 1 } = \frac { 4 } { x ^ { 4 } - 1 }$$

Whakaoti mō x (complex solution)

$$x\in \mathrm{C}\setminus -1,1,-i,i$$

Ngā Upane Otinga

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -1,-i,i,1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-1\right)\left(x+1\right)\left(x-i\right)\left(x+i\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}-1,x^{2}+1,x^{4}-1.

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

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}+1 ki te 2.

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

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-1 ki te 2.

$$2x^{2}+2-\left(2x^{2}-2\right)=4$$

Hei kimi i te tauaro o 2x^{2}-2, kimihia te tauaro o ia taurangi.

$$2x^{2}+2-2x^{2}+2=4$$

Pahekotia te 2x^{2} me -2x^{2}, ka 0.

$$2+2=4$$

Tāpirihia te 2 ki te 2, ka 4.

$$4=4$$

Whakatauritea te 4 me te 4.

$$\text{true}$$

He pono tēnei mō tētahi x ahakoa.

$$x\in \mathrm{C}$$

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -i,i,-1,1.

$$x\in \mathrm{C}\setminus -i,i,-1,1$$

Whakaoti mō x

$$x\in \mathrm{R}\setminus 1,-1$$

Ngā Upane Otinga

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -1,1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-1\right)\left(x+1\right)\left(x^{2}+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}-1,x^{2}+1,x^{4}-1.

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

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}+1 ki te 2.

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

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-1 ki te 2.

$$2x^{2}+2-\left(2x^{2}-2\right)=4$$

Hei kimi i te tauaro o 2x^{2}-2, kimihia te tauaro o ia taurangi.

$$2x^{2}+2-2x^{2}+2=4$$

Pahekotia te 2x^{2} me -2x^{2}, ka 0.

$$2+2=4$$

Tāpirihia te 2 ki te 2, ka 4.

$$4=4$$

Whakatauritea te 4 me te 4.

$$\text{true}$$

He pono tēnei mō tētahi x ahakoa.

$$x\in \mathrm{R}$$

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -1,1.

$$x\in \mathrm{R}\setminus -1,1$$