Whakaoti i te 8n^2-4(1-2n)(2+8n)=0

$$8 n ^ { 2 } - 4 ( 1 - 2 n ) ( 2 + 8 n ) = 0$$

Whakaoti mō n

$$n=\frac{\sqrt{10}+1}{9}\approx 0.462475296$$

$$n=\frac{1-\sqrt{10}}{9}\approx -0.240253073$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia te -1 ki te 4, ka -4.

$$8n^{2}-4\left(1-2n\right)\left(2+8n\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -4 ki te 1-2n.

$$8n^{2}+\left(-4+8n\right)\left(2+8n\right)=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -4+8n ki te 2+8n ka whakakotahi i ngā kupu rite.

$$8n^{2}-8-16n+64n^{2}=0$$

Pahekotia te 8n^{2} me 64n^{2}, ka 72n^{2}.

$$72n^{2}-8-16n=0$$

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.

$$72n^{2}-16n-8=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 72 mō a, -16 mō b, me -8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 72\left(-8\right)}}{2\times 72}$$

Pūrua -16.

$$n=\frac{-\left(-16\right)±\sqrt{256-4\times 72\left(-8\right)}}{2\times 72}$$

Whakareatia -4 ki te 72.

$$n=\frac{-\left(-16\right)±\sqrt{256-288\left(-8\right)}}{2\times 72}$$

Whakareatia -288 ki te -8.

$$n=\frac{-\left(-16\right)±\sqrt{256+2304}}{2\times 72}$$

Tāpiri 256 ki te 2304.

$$n=\frac{-\left(-16\right)±\sqrt{2560}}{2\times 72}$$

Tuhia te pūtakerua o te 2560.

$$n=\frac{-\left(-16\right)±16\sqrt{10}}{2\times 72}$$

Ko te tauaro o -16 ko 16.

$$n=\frac{16±16\sqrt{10}}{2\times 72}$$

Whakareatia 2 ki te 72.

$$n=\frac{16±16\sqrt{10}}{144}$$

Nā, me whakaoti te whārite n=\frac{16±16\sqrt{10}}{144} ina he tāpiri te ±. Tāpiri 16 ki te 16\sqrt{10}.

$$n=\frac{16\sqrt{10}+16}{144}$$

Whakawehe 16+16\sqrt{10} ki te 144.

$$n=\frac{\sqrt{10}+1}{9}$$

Nā, me whakaoti te whārite n=\frac{16±16\sqrt{10}}{144} ina he tango te ±. Tango 16\sqrt{10} mai i 16.

$$n=\frac{16-16\sqrt{10}}{144}$$

Whakawehe 16-16\sqrt{10} ki te 144.

$$n=\frac{1-\sqrt{10}}{9}$$

Kua oti te whārite te whakatau.

$$n=\frac{\sqrt{10}+1}{9} n=\frac{1-\sqrt{10}}{9}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia te -1 ki te 4, ka -4.

$$8n^{2}-4\left(1-2n\right)\left(2+8n\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -4 ki te 1-2n.

$$8n^{2}+\left(-4+8n\right)\left(2+8n\right)=0$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -4+8n ki te 2+8n ka whakakotahi i ngā kupu rite.

$$8n^{2}-8-16n+64n^{2}=0$$

Pahekotia te 8n^{2} me 64n^{2}, ka 72n^{2}.

$$72n^{2}-8-16n=0$$

Me tāpiri te 8 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$72n^{2}-16n=8$$

Whakawehea ngā taha e rua ki te 72.

$$\frac{72n^{2}-16n}{72}=\frac{8}{72}$$

Mā te whakawehe ki te 72 ka wetekia te whakareanga ki te 72.

$$n^{2}+\left(-\frac{16}{72}\right)n=\frac{8}{72}$$

Whakahekea te hautanga \frac{-16}{72} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$n^{2}-\frac{2}{9}n=\frac{8}{72}$$

Whakahekea te hautanga \frac{8}{72} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$n^{2}-\frac{2}{9}n=\frac{1}{9}$$

Whakawehea te -\frac{2}{9}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{9}. Nā, tāpiria te pūrua o te -\frac{1}{9} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia -\frac{1}{9} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}-\frac{2}{9}n+\frac{1}{81}=\frac{1}{9}+\frac{1}{81}$$

Tāpiri \frac{1}{9} ki te \frac{1}{81} 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.

$$n^{2}-\frac{2}{9}n+\frac{1}{81}=\frac{10}{81}$$

Tauwehea n^{2}-\frac{2}{9}n+\frac{1}{81}. 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}.

$$\left(n-\frac{1}{9}\right)^{2}=\frac{10}{81}$$

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

$$\sqrt{\left(n-\frac{1}{9}\right)^{2}}=\sqrt{\frac{10}{81}}$$

Whakarūnātia.

$$n-\frac{1}{9}=\frac{\sqrt{10}}{9} n-\frac{1}{9}=-\frac{\sqrt{10}}{9}$$

Me tāpiri \frac{1}{9} ki ngā taha e rua o te whārite.

$$n=\frac{\sqrt{10}+1}{9} n=\frac{1-\sqrt{10}}{9}$$