Whakaoti i te n[-34+(n-1)*3]/2=230

$$\frac { n [ - 34 + ( n - 1 ) \times 3 ] } { 2 } = 230$$

Whakaoti mō n

$$n = -\frac{23}{3} = -7\frac{2}{3} \approx -7.666666667$$

$$n=20$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me whakarea ngā taha e rua ki te 2.

$$n\left(-34+\left(n-1\right)\times 3\right)=230\times 2$$

Whakamahia te āhuatanga tohatoha hei whakarea te n-1 ki te 3.

$$n\left(-34+3n-3\right)=230\times 2$$

Tangohia te 3 i te -34, ka -37.

$$n\left(-37+3n\right)=230\times 2$$

Whakamahia te āhuatanga tohatoha hei whakarea te n ki te -37+3n.

$$-37n+3n^{2}=230\times 2$$

Whakareatia te 230 ki te 2, ka 460.

$$-37n+3n^{2}=460$$

Tangohia te 460 mai i ngā taha e rua.

$$-37n+3n^{2}-460=0$$

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.

$$3n^{2}-37n-460=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -37 mō b, me -460 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-37\right)±\sqrt{\left(-37\right)^{2}-4\times 3\left(-460\right)}}{2\times 3}$$

Pūrua -37.

$$n=\frac{-\left(-37\right)±\sqrt{1369-4\times 3\left(-460\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$n=\frac{-\left(-37\right)±\sqrt{1369-12\left(-460\right)}}{2\times 3}$$

Whakareatia -12 ki te -460.

$$n=\frac{-\left(-37\right)±\sqrt{1369+5520}}{2\times 3}$$

Tāpiri 1369 ki te 5520.

$$n=\frac{-\left(-37\right)±\sqrt{6889}}{2\times 3}$$

Tuhia te pūtakerua o te 6889.

$$n=\frac{-\left(-37\right)±83}{2\times 3}$$

Ko te tauaro o -37 ko 37.

$$n=\frac{37±83}{2\times 3}$$

Whakareatia 2 ki te 3.

$$n=\frac{37±83}{6}$$

Nā, me whakaoti te whārite n=\frac{37±83}{6} ina he tāpiri te ±. Tāpiri 37 ki te 83.

$$n=\frac{120}{6}$$

Whakawehe 120 ki te 6.

$$n=20$$

Nā, me whakaoti te whārite n=\frac{37±83}{6} ina he tango te ±. Tango 83 mai i 37.

$$n=-\frac{46}{6}$$

Whakahekea te hautanga \frac{-46}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$n=-\frac{23}{3}$$

Kua oti te whārite te whakatau.

$$n=20 n=-\frac{23}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Me whakarea ngā taha e rua ki te 2.

$$n\left(-34+\left(n-1\right)\times 3\right)=230\times 2$$

Whakamahia te āhuatanga tohatoha hei whakarea te n-1 ki te 3.

$$n\left(-34+3n-3\right)=230\times 2$$

Tangohia te 3 i te -34, ka -37.

$$n\left(-37+3n\right)=230\times 2$$

Whakamahia te āhuatanga tohatoha hei whakarea te n ki te -37+3n.

$$-37n+3n^{2}=230\times 2$$

Whakareatia te 230 ki te 2, ka 460.

$$-37n+3n^{2}=460$$

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.

$$3n^{2}-37n=460$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3n^{2}-37n}{3}=\frac{460}{3}$$

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

$$n^{2}-\frac{37}{3}n=\frac{460}{3}$$

Whakawehea te -\frac{37}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{37}{6}. Nā, tāpiria te pūrua o te -\frac{37}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}-\frac{37}{3}n+\left(-\frac{37}{6}\right)^{2}=\frac{460}{3}+\left(-\frac{37}{6}\right)^{2}$$

Pūruatia -\frac{37}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}-\frac{37}{3}n+\frac{1369}{36}=\frac{460}{3}+\frac{1369}{36}$$

Tāpiri \frac{460}{3} ki te \frac{1369}{36} 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.

$$n^{2}-\frac{37}{3}n+\frac{1369}{36}=\frac{6889}{36}$$

Tauwehea n^{2}-\frac{37}{3}n+\frac{1369}{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}.

$$\left(n-\frac{37}{6}\right)^{2}=\frac{6889}{36}$$

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

$$\sqrt{\left(n-\frac{37}{6}\right)^{2}}=\sqrt{\frac{6889}{36}}$$

Whakarūnātia.

$$n-\frac{37}{6}=\frac{83}{6} n-\frac{37}{6}=-\frac{83}{6}$$

Me tāpiri \frac{37}{6} ki ngā taha e rua o te whārite.

$$n=20 n=-\frac{23}{3}$$