Whakaoti i te 9=n-4/3n^2+2/3n^2

$$9 = \frac{ n-4 }{ 3 { n }^{ 2 } } + \frac{ 2 }{ 3 { n }^{ 2 } }$$

Whakaoti mō n

$$n=\frac{1+\sqrt{215}i}{54}\approx 0.018518519+0.271534783i$$

$$n=\frac{-\sqrt{215}i+1}{54}\approx 0.018518519-0.271534783i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe n ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 3n^{2}.

$$27n^{2}=n-4+2$$

Tāpirihia te -4 ki te 2, ka -2.

$$27n^{2}=n-2$$

Tangohia te n mai i ngā taha e rua.

$$27n^{2}-n=-2$$

Me tāpiri te 2 ki ngā taha e rua.

$$27n^{2}-n+2=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 27 mō a, -1 mō b, me 2 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-1\right)±\sqrt{1-4\times 27\times 2}}{2\times 27}$$

Whakareatia -4 ki te 27.

$$n=\frac{-\left(-1\right)±\sqrt{1-108\times 2}}{2\times 27}$$

Whakareatia -108 ki te 2.

$$n=\frac{-\left(-1\right)±\sqrt{1-216}}{2\times 27}$$

Tāpiri 1 ki te -216.

$$n=\frac{-\left(-1\right)±\sqrt{-215}}{2\times 27}$$

Tuhia te pūtakerua o te -215.

$$n=\frac{-\left(-1\right)±\sqrt{215}i}{2\times 27}$$

Ko te tauaro o -1 ko 1.

$$n=\frac{1±\sqrt{215}i}{2\times 27}$$

Whakareatia 2 ki te 27.

$$n=\frac{1±\sqrt{215}i}{54}$$

Nā, me whakaoti te whārite n=\frac{1±\sqrt{215}i}{54} ina he tāpiri te ±. Tāpiri 1 ki te i\sqrt{215}.

$$n=\frac{1+\sqrt{215}i}{54}$$

Nā, me whakaoti te whārite n=\frac{1±\sqrt{215}i}{54} ina he tango te ±. Tango i\sqrt{215} mai i 1.

$$n=\frac{-\sqrt{215}i+1}{54}$$

Kua oti te whārite te whakatau.

$$n=\frac{1+\sqrt{215}i}{54} n=\frac{-\sqrt{215}i+1}{54}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe n ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 3n^{2}.

$$27n^{2}=n-4+2$$

Tāpirihia te -4 ki te 2, ka -2.

$$27n^{2}=n-2$$

Tangohia te n mai i ngā taha e rua.

$$27n^{2}-n=-2$$

Whakawehea ngā taha e rua ki te 27.

$$\frac{27n^{2}-n}{27}=-\frac{2}{27}$$

Mā te whakawehe ki te 27 ka wetekia te whakareanga ki te 27.

$$n^{2}-\frac{1}{27}n=-\frac{2}{27}$$

Whakawehea te -\frac{1}{27}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{54}. Nā, tāpiria te pūrua o te -\frac{1}{54} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}-\frac{1}{27}n+\left(-\frac{1}{54}\right)^{2}=-\frac{2}{27}+\left(-\frac{1}{54}\right)^{2}$$

Pūruatia -\frac{1}{54} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}-\frac{1}{27}n+\frac{1}{2916}=-\frac{2}{27}+\frac{1}{2916}$$

Tāpiri -\frac{2}{27} ki te \frac{1}{2916} 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.

$$n^{2}-\frac{1}{27}n+\frac{1}{2916}=-\frac{215}{2916}$$

Tauwehea n^{2}-\frac{1}{27}n+\frac{1}{2916}. 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}.

$$\left(n-\frac{1}{54}\right)^{2}=-\frac{215}{2916}$$

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

$$\sqrt{\left(n-\frac{1}{54}\right)^{2}}=\sqrt{-\frac{215}{2916}}$$

Whakarūnātia.

$$n-\frac{1}{54}=\frac{\sqrt{215}i}{54} n-\frac{1}{54}=-\frac{\sqrt{215}i}{54}$$

Me tāpiri \frac{1}{54} ki ngā taha e rua o te whārite.

$$n=\frac{1+\sqrt{215}i}{54} n=\frac{-\sqrt{215}i+1}{54}$$