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

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

Whakaoti mō y

$$y=-\frac{2\left(x^{2}-x+16\right)}{x^{2}+x-16}$$

$$x\neq 0\text{ and }x\neq \frac{\sqrt{65}-1}{2}\text{ and }x\neq \frac{-\sqrt{65}-1}{2}\text{ and }x\neq 16$$

Ngā Upane mō te Whakaoti Whārite Rārangi

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

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

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

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

Tātaihia te 4 mā te pū o 2, kia riro ko 16.

$$yx^{2}+2x^{2}=\left(y-2\right)\left(16-x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te y-2 ki te 16-x.

$$yx^{2}+2x^{2}=16y-yx-32+2x$$

Tangohia te 16y mai i ngā taha e rua.

$$yx^{2}+2x^{2}-16y=-yx-32+2x$$

Me tāpiri te yx ki ngā taha e rua.

$$yx^{2}+2x^{2}-16y+yx=-32+2x$$

Tangohia te 2x^{2} mai i ngā taha e rua.

$$yx^{2}-16y+yx=-32+2x-2x^{2}$$

Pahekotia ngā kīanga tau katoa e whai ana i te y.

$$\left(x^{2}-16+x\right)y=-32+2x-2x^{2}$$

He hanga arowhānui tō te whārite.

$$\left(x^{2}+x-16\right)y=-2x^{2}+2x-32$$

Whakawehea ngā taha e rua ki te x^{2}-16+x.

$$\frac{\left(x^{2}+x-16\right)y}{x^{2}+x-16}=\frac{-2x^{2}+2x-32}{x^{2}+x-16}$$

Mā te whakawehe ki te x^{2}-16+x ka wetekia te whakareanga ki te x^{2}-16+x.

$$y=\frac{-2x^{2}+2x-32}{x^{2}+x-16}$$

Whakawehe -32+2x-2x^{2} ki te x^{2}-16+x.

$$y=\frac{2\left(-x^{2}+x-16\right)}{x^{2}+x-16}$$

Tē taea kia ōrite te tāupe y ki tētahi o ngā uara -2,2.

$$y=\frac{2\left(-x^{2}+x-16\right)}{x^{2}+x-16}\text{, }y\neq -2\text{ and }y\neq 2$$

Whakaoti mō x (complex solution)

$$x=\frac{\sqrt{\left(y-2\right)\left(65y+126\right)}-y+2}{2\left(y+2\right)}$$

$$x=\frac{-\sqrt{\left(y-2\right)\left(65y+126\right)}-y+2}{2\left(y+2\right)}\text{, }y\neq 2\text{ and }y\neq -2$$

Whakaoti mō x

$$x=\frac{\sqrt{\left(y-2\right)\left(65y+126\right)}-y+2}{2\left(y+2\right)}$$

$$x=\frac{-\sqrt{\left(y-2\right)\left(65y+126\right)}-y+2}{2\left(y+2\right)}\text{, }y>2\text{ or }\left(y\neq -2\text{ and }y\leq -\frac{126}{65}\right)$$