Whakaoti i te 8n^2+4n-16=-n^2

$$8 n ^ { 2 } + 4 n - 16 = - n ^ { 2 }$$

Whakaoti mō n

$$n = \frac{2 \sqrt{37} - 2}{9} \approx 1.129502785$$

$$n=\frac{-2\sqrt{37}-2}{9}\approx -1.573947229$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me tāpiri te n^{2} ki ngā taha e rua.

$$8n^{2}+4n-16+n^{2}=0$$

Pahekotia te 8n^{2} me n^{2}, ka 9n^{2}.

$$9n^{2}+4n-16=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, 4 mō b, me -16 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-4±\sqrt{4^{2}-4\times 9\left(-16\right)}}{2\times 9}$$

Pūrua 4.

$$n=\frac{-4±\sqrt{16-4\times 9\left(-16\right)}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$n=\frac{-4±\sqrt{16-36\left(-16\right)}}{2\times 9}$$

Whakareatia -36 ki te -16.

$$n=\frac{-4±\sqrt{16+576}}{2\times 9}$$

Tāpiri 16 ki te 576.

$$n=\frac{-4±\sqrt{592}}{2\times 9}$$

Tuhia te pūtakerua o te 592.

$$n=\frac{-4±4\sqrt{37}}{2\times 9}$$

Whakareatia 2 ki te 9.

$$n=\frac{-4±4\sqrt{37}}{18}$$

Nā, me whakaoti te whārite n=\frac{-4±4\sqrt{37}}{18} ina he tāpiri te ±. Tāpiri -4 ki te 4\sqrt{37}.

$$n=\frac{4\sqrt{37}-4}{18}$$

Whakawehe -4+4\sqrt{37} ki te 18.

$$n=\frac{2\sqrt{37}-2}{9}$$

Nā, me whakaoti te whārite n=\frac{-4±4\sqrt{37}}{18} ina he tango te ±. Tango 4\sqrt{37} mai i -4.

$$n=\frac{-4\sqrt{37}-4}{18}$$

Whakawehe -4-4\sqrt{37} ki te 18.

$$n=\frac{-2\sqrt{37}-2}{9}$$

Kua oti te whārite te whakatau.

$$n=\frac{2\sqrt{37}-2}{9} n=\frac{-2\sqrt{37}-2}{9}$$

Ngā Upane mō te Whakaoti i te Pūrua

Me tāpiri te n^{2} ki ngā taha e rua.

$$8n^{2}+4n-16+n^{2}=0$$

Pahekotia te 8n^{2} me n^{2}, ka 9n^{2}.

$$9n^{2}+4n-16=0$$

Me tāpiri te 16 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$9n^{2}+4n=16$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9n^{2}+4n}{9}=\frac{16}{9}$$

Mā te whakawehe ki te 9 ka wetekia te whakareanga ki te 9.

$$n^{2}+\frac{4}{9}n=\frac{16}{9}$$

Whakawehea te \frac{4}{9}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{2}{9}. Nā, tāpiria te pūrua o te \frac{2}{9} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}+\frac{4}{9}n+\left(\frac{2}{9}\right)^{2}=\frac{16}{9}+\left(\frac{2}{9}\right)^{2}$$

Pūruatia \frac{2}{9} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}+\frac{4}{9}n+\frac{4}{81}=\frac{16}{9}+\frac{4}{81}$$

Tāpiri \frac{16}{9} ki te \frac{4}{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.

$$n^{2}+\frac{4}{9}n+\frac{4}{81}=\frac{148}{81}$$

Tauwehea n^{2}+\frac{4}{9}n+\frac{4}{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}.

$$\left(n+\frac{2}{9}\right)^{2}=\frac{148}{81}$$

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

$$\sqrt{\left(n+\frac{2}{9}\right)^{2}}=\sqrt{\frac{148}{81}}$$

Whakarūnātia.

$$n+\frac{2}{9}=\frac{2\sqrt{37}}{9} n+\frac{2}{9}=-\frac{2\sqrt{37}}{9}$$

Me tango \frac{2}{9} mai i ngā taha e rua o te whārite.

$$n=\frac{2\sqrt{37}-2}{9} n=\frac{-2\sqrt{37}-2}{9}$$