Whakaoti i te 3n^2+47n-232=5

$$3 n ^ { 2 } + 47 n - 232 = 5$$

Whakaoti mō n

$$n = \frac{\sqrt{5053} - 47}{6} \approx 4.014076135$$

$$n=\frac{-\sqrt{5053}-47}{6}\approx -19.680742802$$

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.

$$3n^{2}+47n-232=5$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$3n^{2}+47n-232-5=5-5$$

Mā te tango i te 5 i a ia ake anō ka toe ko te 0.

$$3n^{2}+47n-232-5=0$$

Tango 5 mai i -232.

$$3n^{2}+47n-237=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 47 mō b, me -237 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-47±\sqrt{47^{2}-4\times 3\left(-237\right)}}{2\times 3}$$

Pūrua 47.

$$n=\frac{-47±\sqrt{2209-4\times 3\left(-237\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$n=\frac{-47±\sqrt{2209-12\left(-237\right)}}{2\times 3}$$

Whakareatia -12 ki te -237.

$$n=\frac{-47±\sqrt{2209+2844}}{2\times 3}$$

Tāpiri 2209 ki te 2844.

$$n=\frac{-47±\sqrt{5053}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$n=\frac{-47±\sqrt{5053}}{6}$$

Nā, me whakaoti te whārite n=\frac{-47±\sqrt{5053}}{6} ina he tāpiri te ±. Tāpiri -47 ki te \sqrt{5053}.

$$n=\frac{\sqrt{5053}-47}{6}$$

Nā, me whakaoti te whārite n=\frac{-47±\sqrt{5053}}{6} ina he tango te ±. Tango \sqrt{5053} mai i -47.

$$n=\frac{-\sqrt{5053}-47}{6}$$

Kua oti te whārite te whakatau.

$$n=\frac{\sqrt{5053}-47}{6} n=\frac{-\sqrt{5053}-47}{6}$$

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.

$$3n^{2}+47n-232=5$$

Me tāpiri 232 ki ngā taha e rua o te whārite.

$$3n^{2}+47n-232-\left(-232\right)=5-\left(-232\right)$$

Mā te tango i te -232 i a ia ake anō ka toe ko te 0.

$$3n^{2}+47n=5-\left(-232\right)$$

Tango -232 mai i 5.

$$3n^{2}+47n=237$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3n^{2}+47n}{3}=\frac{237}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$n^{2}+\frac{47}{3}n=\frac{237}{3}$$

Whakawehe 237 ki te 3.

$$n^{2}+\frac{47}{3}n=79$$

Whakawehea te \frac{47}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{47}{6}. Nā, tāpiria te pūrua o te \frac{47}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}+\frac{47}{3}n+\left(\frac{47}{6}\right)^{2}=79+\left(\frac{47}{6}\right)^{2}$$

Pūruatia \frac{47}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}+\frac{47}{3}n+\frac{2209}{36}=79+\frac{2209}{36}$$

Tāpiri 79 ki te \frac{2209}{36}.

$$n^{2}+\frac{47}{3}n+\frac{2209}{36}=\frac{5053}{36}$$

Tauwehea n^{2}+\frac{47}{3}n+\frac{2209}{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}.

$$\left(n+\frac{47}{6}\right)^{2}=\frac{5053}{36}$$

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

$$\sqrt{\left(n+\frac{47}{6}\right)^{2}}=\sqrt{\frac{5053}{36}}$$

Whakarūnātia.

$$n+\frac{47}{6}=\frac{\sqrt{5053}}{6} n+\frac{47}{6}=-\frac{\sqrt{5053}}{6}$$

Me tango \frac{47}{6} mai i ngā taha e rua o te whārite.

$$n=\frac{\sqrt{5053}-47}{6} n=\frac{-\sqrt{5053}-47}{6}$$