Whakaoti i te x/x-1=8x+1/x-1

$$\frac { x } { x - 1 } = 8 x + \frac { 1 } { x - 1 }$$

Whakaoti mō x

$$x=\frac{1}{8}=0.125$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$x=8x\left(x-1\right)+1$$

Whakamahia te āhuatanga tohatoha hei whakarea te 8x ki te x-1.

$$x=8x^{2}-8x+1$$

Tangohia te 8x^{2} mai i ngā taha e rua.

$$x-8x^{2}=-8x+1$$

Me tāpiri te 8x ki ngā taha e rua.

$$x-8x^{2}+8x=1$$

Pahekotia te x me 8x, ka 9x.

$$9x-8x^{2}=1$$

Tangohia te 1 mai i ngā taha e rua.

$$9x-8x^{2}-1=0$$

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.

$$-8x^{2}+9x-1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -8 mō a, 9 mō b, me -1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-8\right)\left(-1\right)}}{2\left(-8\right)}$$

Pūrua 9.

$$x=\frac{-9±\sqrt{81-4\left(-8\right)\left(-1\right)}}{2\left(-8\right)}$$

Whakareatia -4 ki te -8.

$$x=\frac{-9±\sqrt{81+32\left(-1\right)}}{2\left(-8\right)}$$

Whakareatia 32 ki te -1.

$$x=\frac{-9±\sqrt{81-32}}{2\left(-8\right)}$$

Tāpiri 81 ki te -32.

$$x=\frac{-9±\sqrt{49}}{2\left(-8\right)}$$

Tuhia te pūtakerua o te 49.

$$x=\frac{-9±7}{2\left(-8\right)}$$

Whakareatia 2 ki te -8.

$$x=\frac{-9±7}{-16}$$

Nā, me whakaoti te whārite x=\frac{-9±7}{-16} ina he tāpiri te ±. Tāpiri -9 ki te 7.

$$x=-\frac{2}{-16}$$

Whakahekea te hautanga \frac{-2}{-16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{1}{8}$$

Nā, me whakaoti te whārite x=\frac{-9±7}{-16} ina he tango te ±. Tango 7 mai i -9.

$$x=-\frac{16}{-16}$$

Whakawehe -16 ki te -16.

$$x=1$$

Kua oti te whārite te whakatau.

$$x=\frac{1}{8} x=1$$

Tē taea kia ōrite te tāupe x ki 1.

$$x=\frac{1}{8}$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$x=8x\left(x-1\right)+1$$

Whakamahia te āhuatanga tohatoha hei whakarea te 8x ki te x-1.

$$x=8x^{2}-8x+1$$

Tangohia te 8x^{2} mai i ngā taha e rua.

$$x-8x^{2}=-8x+1$$

Me tāpiri te 8x ki ngā taha e rua.

$$x-8x^{2}+8x=1$$

Pahekotia te x me 8x, ka 9x.

$$9x-8x^{2}=1$$

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.

$$-8x^{2}+9x=1$$

Whakawehea ngā taha e rua ki te -8.

$$\frac{-8x^{2}+9x}{-8}=\frac{1}{-8}$$

Mā te whakawehe ki te -8 ka wetekia te whakareanga ki te -8.

$$x^{2}+\frac{9}{-8}x=\frac{1}{-8}$$

Whakawehe 9 ki te -8.

$$x^{2}-\frac{9}{8}x=\frac{1}{-8}$$

Whakawehe 1 ki te -8.

$$x^{2}-\frac{9}{8}x=-\frac{1}{8}$$

Whakawehea te -\frac{9}{8}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{16}. Nā, tāpiria te pūrua o te -\frac{9}{16} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{9}{8}x+\left(-\frac{9}{16}\right)^{2}=-\frac{1}{8}+\left(-\frac{9}{16}\right)^{2}$$

Pūruatia -\frac{9}{16} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{9}{8}x+\frac{81}{256}=-\frac{1}{8}+\frac{81}{256}$$

Tāpiri -\frac{1}{8} ki te \frac{81}{256} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-\frac{9}{8}x+\frac{81}{256}=\frac{49}{256}$$

Tauwehea x^{2}-\frac{9}{8}x+\frac{81}{256}. 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}.

$$\left(x-\frac{9}{16}\right)^{2}=\frac{49}{256}$$

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

$$\sqrt{\left(x-\frac{9}{16}\right)^{2}}=\sqrt{\frac{49}{256}}$$

Whakarūnātia.

$$x-\frac{9}{16}=\frac{7}{16} x-\frac{9}{16}=-\frac{7}{16}$$

Me tāpiri \frac{9}{16} ki ngā taha e rua o te whārite.

$$x=1 x=\frac{1}{8}$$

Tē taea kia ōrite te tāupe x ki 1.

$$x=\frac{1}{8}$$