Whakaoti i te 9n^2-23n+20=3n^2

$$9 n ^ { 2 } - 23 n + 20 = 3 n ^ { 2 }$$

Whakaoti mō n

$$n = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

$$n = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 3n^{2} mai i ngā taha e rua.

$$9n^{2}-23n+20-3n^{2}=0$$

Pahekotia te 9n^{2} me -3n^{2}, ka 6n^{2}.

$$6n^{2}-23n+20=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 6n^{2}+an+bn+20. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-23 ab=6\times 20=120$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 120.

$$-1,-120 -2,-60 -3,-40 -4,-30 -5,-24 -6,-20 -8,-15 -10,-12$$

Tātaihia te tapeke mō ia takirua.

$$-1-120=-121 -2-60=-62 -3-40=-43 -4-30=-34 -5-24=-29 -6-20=-26 -8-15=-23 -10-12=-22$$

Ko te otinga te takirua ka hoatu i te tapeke -23.

$$a=-15 b=-8$$

Tuhia anō te 6n^{2}-23n+20 hei \left(6n^{2}-15n\right)+\left(-8n+20\right).

$$\left(6n^{2}-15n\right)+\left(-8n+20\right)$$

Tauwehea te 3n i te tuatahi me te -4 i te rōpū tuarua.

$$3n\left(2n-5\right)-4\left(2n-5\right)$$

Whakatauwehea atu te kīanga pātahi 2n-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2n-5\right)\left(3n-4\right)$$

Hei kimi otinga whārite, me whakaoti te 2n-5=0 me te 3n-4=0.

$$n=\frac{5}{2} n=\frac{4}{3}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 3n^{2} mai i ngā taha e rua.

$$9n^{2}-23n+20-3n^{2}=0$$

Pahekotia te 9n^{2} me -3n^{2}, ka 6n^{2}.

$$6n^{2}-23n+20=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, -23 mō b, me 20 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-23\right)±\sqrt{\left(-23\right)^{2}-4\times 6\times 20}}{2\times 6}$$

Pūrua -23.

$$n=\frac{-\left(-23\right)±\sqrt{529-4\times 6\times 20}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$n=\frac{-\left(-23\right)±\sqrt{529-24\times 20}}{2\times 6}$$

Whakareatia -24 ki te 20.

$$n=\frac{-\left(-23\right)±\sqrt{529-480}}{2\times 6}$$

Tāpiri 529 ki te -480.

$$n=\frac{-\left(-23\right)±\sqrt{49}}{2\times 6}$$

Tuhia te pūtakerua o te 49.

$$n=\frac{-\left(-23\right)±7}{2\times 6}$$

Ko te tauaro o -23 ko 23.

$$n=\frac{23±7}{2\times 6}$$

Whakareatia 2 ki te 6.

$$n=\frac{23±7}{12}$$

Nā, me whakaoti te whārite n=\frac{23±7}{12} ina he tāpiri te ±. Tāpiri 23 ki te 7.

$$n=\frac{30}{12}$$

Whakahekea te hautanga \frac{30}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$n=\frac{5}{2}$$

Nā, me whakaoti te whārite n=\frac{23±7}{12} ina he tango te ±. Tango 7 mai i 23.

$$n=\frac{16}{12}$$

Whakahekea te hautanga \frac{16}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$n=\frac{4}{3}$$

Kua oti te whārite te whakatau.

$$n=\frac{5}{2} n=\frac{4}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 3n^{2} mai i ngā taha e rua.

$$9n^{2}-23n+20-3n^{2}=0$$

Pahekotia te 9n^{2} me -3n^{2}, ka 6n^{2}.

$$6n^{2}-23n+20=0$$

Tangohia te 20 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$6n^{2}-23n=-20$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6n^{2}-23n}{6}=-\frac{20}{6}$$

Mā te whakawehe ki te 6 ka wetekia te whakareanga ki te 6.

$$n^{2}-\frac{23}{6}n=-\frac{20}{6}$$

Whakahekea te hautanga \frac{-20}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$n^{2}-\frac{23}{6}n=-\frac{10}{3}$$

Whakawehea te -\frac{23}{6}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{23}{12}. Nā, tāpiria te pūrua o te -\frac{23}{12} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}-\frac{23}{6}n+\left(-\frac{23}{12}\right)^{2}=-\frac{10}{3}+\left(-\frac{23}{12}\right)^{2}$$

Pūruatia -\frac{23}{12} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}-\frac{23}{6}n+\frac{529}{144}=-\frac{10}{3}+\frac{529}{144}$$

Tāpiri -\frac{10}{3} ki te \frac{529}{144} 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.

$$n^{2}-\frac{23}{6}n+\frac{529}{144}=\frac{49}{144}$$

Tauwehea te n^{2}-\frac{23}{6}n+\frac{529}{144}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{23}{12}\right)^{2}=\frac{49}{144}$$

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

$$\sqrt{\left(n-\frac{23}{12}\right)^{2}}=\sqrt{\frac{49}{144}}$$

Whakarūnātia.

$$n-\frac{23}{12}=\frac{7}{12} n-\frac{23}{12}=-\frac{7}{12}$$

Me tāpiri \frac{23}{12} ki ngā taha e rua o te whārite.

$$n=\frac{5}{2} n=\frac{4}{3}$$