Whakaoti i te 1/t+1/t-1=2

$$\frac { 1 } { t } + \frac { 1 } { t - 1 } = 2$$

Whakaoti mō t

$$t=\frac{\sqrt{2}}{2}+1\approx 1.707106781$$

$$t=-\frac{\sqrt{2}}{2}+1\approx 0.292893219$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe t ki tētahi o ngā uara 0,1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te t\left(t-1\right), arā, te tauraro pātahi he tino iti rawa te kitea o t,t-1.

$$t-1+t=2t\left(t-1\right)$$

Pahekotia te t me t, ka 2t.

$$2t-1=2t\left(t-1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2t ki te t-1.

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

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

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

Me tāpiri te 2t ki ngā taha e rua.

$$2t-1-2t^{2}+2t=0$$

Pahekotia te 2t me 2t, ka 4t.

$$4t-1-2t^{2}=0$$

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$-2t^{2}+4t-1=0$$

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

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

Pūrua 4.

$$t=\frac{-4±\sqrt{16-4\left(-2\right)\left(-1\right)}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$t=\frac{-4±\sqrt{16+8\left(-1\right)}}{2\left(-2\right)}$$

Whakareatia 8 ki te -1.

$$t=\frac{-4±\sqrt{16-8}}{2\left(-2\right)}$$

Tāpiri 16 ki te -8.

$$t=\frac{-4±\sqrt{8}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 8.

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

Whakareatia 2 ki te -2.

$$t=\frac{-4±2\sqrt{2}}{-4}$$

Nā, me whakaoti te whārite t=\frac{-4±2\sqrt{2}}{-4} ina he tāpiri te ±. Tāpiri -4 ki te 2\sqrt{2}.

$$t=\frac{2\sqrt{2}-4}{-4}$$

Whakawehe -4+2\sqrt{2} ki te -4.

$$t=-\frac{\sqrt{2}}{2}+1$$

Nā, me whakaoti te whārite t=\frac{-4±2\sqrt{2}}{-4} ina he tango te ±. Tango 2\sqrt{2} mai i -4.

$$t=\frac{-2\sqrt{2}-4}{-4}$$

Whakawehe -4-2\sqrt{2} ki te -4.

$$t=\frac{\sqrt{2}}{2}+1$$

Kua oti te whārite te whakatau.

$$t=-\frac{\sqrt{2}}{2}+1 t=\frac{\sqrt{2}}{2}+1$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe t ki tētahi o ngā uara 0,1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te t\left(t-1\right), arā, te tauraro pātahi he tino iti rawa te kitea o t,t-1.

$$t-1+t=2t\left(t-1\right)$$

Pahekotia te t me t, ka 2t.

$$2t-1=2t\left(t-1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2t ki te t-1.

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

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

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

Me tāpiri te 2t ki ngā taha e rua.

$$2t-1-2t^{2}+2t=0$$

Pahekotia te 2t me 2t, ka 4t.

$$4t-1-2t^{2}=0$$

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

$$4t-2t^{2}=1$$

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$-2t^{2}+4t=1$$

Whakawehea ngā taha e rua ki te -2.

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

Mā te whakawehe ki te -2 ka wetekia te whakareanga ki te -2.

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

Whakawehe 4 ki te -2.

$$t^{2}-2t=\frac{1}{-2}$$

Whakawehe 1 ki te -2.

$$t^{2}-2t=-\frac{1}{2}$$

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.

$$t^{2}-2t+1=-\frac{1}{2}+1$$

Tāpiri -\frac{1}{2} ki te 1.

$$t^{2}-2t+1=\frac{1}{2}$$

Tauwehea t^{2}-2t+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}.

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

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

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

Whakarūnātia.

$$t-1=\frac{\sqrt{2}}{2} t-1=-\frac{\sqrt{2}}{2}$$

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

$$t=\frac{\sqrt{2}}{2}+1 t=-\frac{\sqrt{2}}{2}+1$$