Whakaoti i te 3n^2+10n=8

$$3 n ^ { 2 } + 10 n = 8$$

Whakaoti mō n

$$n=-4$$

$$n=\frac{2}{3}\approx 0.666666667$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 8 mai i ngā taha e rua.

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

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 3n^{2}+an+bn-8. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=10 ab=3\left(-8\right)=-24$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -24.

$$-1,24 -2,12 -3,8 -4,6$$

Tātaihia te tapeke mō ia takirua.

$$-1+24=23 -2+12=10 -3+8=5 -4+6=2$$

Ko te otinga te takirua ka hoatu i te tapeke 10.

$$a=-2 b=12$$

Tuhia anō te 3n^{2}+10n-8 hei \left(3n^{2}-2n\right)+\left(12n-8\right).

$$\left(3n^{2}-2n\right)+\left(12n-8\right)$$

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

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

Whakatauwehea atu te kīanga pātahi 3n-2 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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

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

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.

$$3n^{2}+10n=8$$

Me tango 8 mai i ngā taha e rua o te whārite.

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

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

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

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 10 mō b, me -8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-10±\sqrt{10^{2}-4\times 3\left(-8\right)}}{2\times 3}$$

Pūrua 10.

$$n=\frac{-10±\sqrt{100-4\times 3\left(-8\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$n=\frac{-10±\sqrt{100-12\left(-8\right)}}{2\times 3}$$

Whakareatia -12 ki te -8.

$$n=\frac{-10±\sqrt{100+96}}{2\times 3}$$

Tāpiri 100 ki te 96.

$$n=\frac{-10±\sqrt{196}}{2\times 3}$$

Tuhia te pūtakerua o te 196.

$$n=\frac{-10±14}{2\times 3}$$

Whakareatia 2 ki te 3.

$$n=\frac{-10±14}{6}$$

Nā, me whakaoti te whārite n=\frac{-10±14}{6} ina he tāpiri te ±. Tāpiri -10 ki te 14.

$$n=\frac{4}{6}$$

Whakahekea te hautanga \frac{4}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Nā, me whakaoti te whārite n=\frac{-10±14}{6} ina he tango te ±. Tango 14 mai i -10.

$$n=-\frac{24}{6}$$

Whakawehe -24 ki te 6.

$$n=-4$$

Kua oti te whārite te whakatau.

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

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.

$$3n^{2}+10n=8$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3n^{2}+10n}{3}=\frac{8}{3}$$

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

$$n^{2}+\frac{10}{3}n=\frac{8}{3}$$

Whakawehea te \frac{10}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{3}. Nā, tāpiria te pūrua o te \frac{5}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}+\frac{10}{3}n+\left(\frac{5}{3}\right)^{2}=\frac{8}{3}+\left(\frac{5}{3}\right)^{2}$$

Pūruatia \frac{5}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}+\frac{10}{3}n+\frac{25}{9}=\frac{8}{3}+\frac{25}{9}$$

Tāpiri \frac{8}{3} ki te \frac{25}{9} 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.

$$n^{2}+\frac{10}{3}n+\frac{25}{9}=\frac{49}{9}$$

Tauwehea n^{2}+\frac{10}{3}n+\frac{25}{9}. 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}.

$$\left(n+\frac{5}{3}\right)^{2}=\frac{49}{9}$$

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

$$\sqrt{\left(n+\frac{5}{3}\right)^{2}}=\sqrt{\frac{49}{9}}$$

Whakarūnātia.

$$n+\frac{5}{3}=\frac{7}{3} n+\frac{5}{3}=-\frac{7}{3}$$

Me tango \frac{5}{3} mai i ngā taha e rua o te whārite.

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