Whakaoti i te x=x+1/x-1

$$x = \frac { x + 1 } { x - 1 }$$

Whakaoti mō x

$$x=\sqrt{2}+1\approx 2.414213562$$

$$x=1-\sqrt{2}\approx -0.414213562$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te \frac{x+1}{x-1} mai i ngā taha e rua.

$$x-\frac{x+1}{x-1}=0$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia x ki te \frac{x-1}{x-1}.

$$\frac{x\left(x-1\right)}{x-1}-\frac{x+1}{x-1}=0$$

Tā te mea he rite te tauraro o \frac{x\left(x-1\right)}{x-1} me \frac{x+1}{x-1}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{x\left(x-1\right)-\left(x+1\right)}{x-1}=0$$

Mahia ngā whakarea i roto o x\left(x-1\right)-\left(x+1\right).

$$\frac{x^{2}-x-x-1}{x-1}=0$$

Whakakotahitia ngā kupu rite i x^{2}-x-x-1.

$$\frac{x^{2}-2x-1}{x-1}=0$$

Tē taea kia ōrite te tāupe x ki 1 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x-1.

$$x^{2}-2x-1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -2 mō b, me -1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua -2.

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

Whakareatia -4 ki te -1.

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

Tāpiri 4 ki te 4.

$$x=\frac{-\left(-2\right)±\sqrt{8}}{2}$$

Tuhia te pūtakerua o te 8.

$$x=\frac{-\left(-2\right)±2\sqrt{2}}{2}$$

Ko te tauaro o -2 ko 2.

$$x=\frac{2±2\sqrt{2}}{2}$$

Nā, me whakaoti te whārite x=\frac{2±2\sqrt{2}}{2} ina he tāpiri te ±. Tāpiri 2 ki te 2\sqrt{2}.

$$x=\frac{2\sqrt{2}+2}{2}$$

Whakawehe 2+2\sqrt{2} ki te 2.

$$x=\sqrt{2}+1$$

Nā, me whakaoti te whārite x=\frac{2±2\sqrt{2}}{2} ina he tango te ±. Tango 2\sqrt{2} mai i 2.

$$x=\frac{2-2\sqrt{2}}{2}$$

Whakawehe 2-2\sqrt{2} ki te 2.

$$x=1-\sqrt{2}$$

Kua oti te whārite te whakatau.

$$x=\sqrt{2}+1 x=1-\sqrt{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te \frac{x+1}{x-1} mai i ngā taha e rua.

$$x-\frac{x+1}{x-1}=0$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia x ki te \frac{x-1}{x-1}.

$$\frac{x\left(x-1\right)}{x-1}-\frac{x+1}{x-1}=0$$

Tā te mea he rite te tauraro o \frac{x\left(x-1\right)}{x-1} me \frac{x+1}{x-1}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{x\left(x-1\right)-\left(x+1\right)}{x-1}=0$$

Mahia ngā whakarea i roto o x\left(x-1\right)-\left(x+1\right).

$$\frac{x^{2}-x-x-1}{x-1}=0$$

Whakakotahitia ngā kupu rite i x^{2}-x-x-1.

$$\frac{x^{2}-2x-1}{x-1}=0$$

Tē taea kia ōrite te tāupe x ki 1 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x-1.

$$x^{2}-2x-1=0$$

Me tāpiri te 1 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$x^{2}-2x=1$$

Whakawehea te -2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -1. Nā, tāpiria te pūrua o te -1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-2x+1=1+1$$

Tāpiri 1 ki te 1.

$$x^{2}-2x+1=2$$

Tauwehea x^{2}-2x+1. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

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

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{2}$$

Whakarūnātia.

$$x-1=\sqrt{2} x-1=-\sqrt{2}$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$x=\sqrt{2}+1 x=1-\sqrt{2}$$