Whakaoti i te 9n^2-3n-8=10

$$9 n ^ { 2 } - 3 n - 8 = 10$$

Whakaoti mō n

$$n = \frac{\sqrt{73} + 1}{6} \approx 1.590667291$$

$$n=\frac{1-\sqrt{73}}{6}\approx -1.257333958$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$9n^{2}-3n-8=10$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$9n^{2}-3n-8-10=10-10$$

Mā te tango i te 10 i a ia ake anō ka toe ko te 0.

$$9n^{2}-3n-8-10=0$$

Tango 10 mai i -8.

$$9n^{2}-3n-18=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, -3 mō b, me -18 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 9\left(-18\right)}}{2\times 9}$$

Pūrua -3.

$$n=\frac{-\left(-3\right)±\sqrt{9-4\times 9\left(-18\right)}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$n=\frac{-\left(-3\right)±\sqrt{9-36\left(-18\right)}}{2\times 9}$$

Whakareatia -36 ki te -18.

$$n=\frac{-\left(-3\right)±\sqrt{9+648}}{2\times 9}$$

Tāpiri 9 ki te 648.

$$n=\frac{-\left(-3\right)±\sqrt{657}}{2\times 9}$$

Tuhia te pūtakerua o te 657.

$$n=\frac{-\left(-3\right)±3\sqrt{73}}{2\times 9}$$

Ko te tauaro o -3 ko 3.

$$n=\frac{3±3\sqrt{73}}{2\times 9}$$

Whakareatia 2 ki te 9.

$$n=\frac{3±3\sqrt{73}}{18}$$

Nā, me whakaoti te whārite n=\frac{3±3\sqrt{73}}{18} ina he tāpiri te ±. Tāpiri 3 ki te 3\sqrt{73}.

$$n=\frac{3\sqrt{73}+3}{18}$$

Whakawehe 3+3\sqrt{73} ki te 18.

$$n=\frac{\sqrt{73}+1}{6}$$

Nā, me whakaoti te whārite n=\frac{3±3\sqrt{73}}{18} ina he tango te ±. Tango 3\sqrt{73} mai i 3.

$$n=\frac{3-3\sqrt{73}}{18}$$

Whakawehe 3-3\sqrt{73} ki te 18.

$$n=\frac{1-\sqrt{73}}{6}$$

Kua oti te whārite te whakatau.

$$n=\frac{\sqrt{73}+1}{6} n=\frac{1-\sqrt{73}}{6}$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$9n^{2}-3n-8=10$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$9n^{2}-3n-8-\left(-8\right)=10-\left(-8\right)$$

Mā te tango i te -8 i a ia ake anō ka toe ko te 0.

$$9n^{2}-3n=10-\left(-8\right)$$

Tango -8 mai i 10.

$$9n^{2}-3n=18$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9n^{2}-3n}{9}=\frac{18}{9}$$

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

$$n^{2}+\left(-\frac{3}{9}\right)n=\frac{18}{9}$$

Whakahekea te hautanga \frac{-3}{9} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$n^{2}-\frac{1}{3}n=\frac{18}{9}$$

Whakawehe 18 ki te 9.

$$n^{2}-\frac{1}{3}n=2$$

Whakawehea te -\frac{1}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{6}. Nā, tāpiria te pūrua o te -\frac{1}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}-\frac{1}{3}n+\left(-\frac{1}{6}\right)^{2}=2+\left(-\frac{1}{6}\right)^{2}$$

Pūruatia -\frac{1}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}-\frac{1}{3}n+\frac{1}{36}=2+\frac{1}{36}$$

Tāpiri 2 ki te \frac{1}{36}.

$$n^{2}-\frac{1}{3}n+\frac{1}{36}=\frac{73}{36}$$

Tauwehea n^{2}-\frac{1}{3}n+\frac{1}{36}. 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}.

$$\left(n-\frac{1}{6}\right)^{2}=\frac{73}{36}$$

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

$$\sqrt{\left(n-\frac{1}{6}\right)^{2}}=\sqrt{\frac{73}{36}}$$

Whakarūnātia.

$$n-\frac{1}{6}=\frac{\sqrt{73}}{6} n-\frac{1}{6}=-\frac{\sqrt{73}}{6}$$

Me tāpiri \frac{1}{6} ki ngā taha e rua o te whārite.

$$n=\frac{\sqrt{73}+1}{6} n=\frac{1-\sqrt{73}}{6}$$