Whakaoti i te 3n^2-4n-15=0

$$3 n ^ { 2 } - 4 n - 15 = 0$$

Whakaoti mō n

$$n = -\frac{5}{3} = -1\frac{2}{3} \approx -1.666666667$$

$$n=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 3n^{2}+an+bn-15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-4 ab=3\left(-15\right)=-45$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -45.

$$1,-45 3,-15 5,-9$$

Tātaihia te tapeke mō ia takirua.

$$1-45=-44 3-15=-12 5-9=-4$$

Ko te otinga te takirua ka hoatu i te tapeke -4.

$$a=-9 b=5$$

Tuhia anō te 3n^{2}-4n-15 hei \left(3n^{2}-9n\right)+\left(5n-15\right).

$$\left(3n^{2}-9n\right)+\left(5n-15\right)$$

Tauwehea te 3n i te tuatahi me te 5 i te rōpū tuarua.

$$3n\left(n-3\right)+5\left(n-3\right)$$

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

$$\left(n-3\right)\left(3n+5\right)$$

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

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

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.

$$3n^{2}-4n-15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -4 mō b, me -15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 3\left(-15\right)}}{2\times 3}$$

Pūrua -4.

$$n=\frac{-\left(-4\right)±\sqrt{16-4\times 3\left(-15\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$n=\frac{-\left(-4\right)±\sqrt{16-12\left(-15\right)}}{2\times 3}$$

Whakareatia -12 ki te -15.

$$n=\frac{-\left(-4\right)±\sqrt{16+180}}{2\times 3}$$

Tāpiri 16 ki te 180.

$$n=\frac{-\left(-4\right)±\sqrt{196}}{2\times 3}$$

Tuhia te pūtakerua o te 196.

$$n=\frac{-\left(-4\right)±14}{2\times 3}$$

Ko te tauaro o -4 ko 4.

$$n=\frac{4±14}{2\times 3}$$

Whakareatia 2 ki te 3.

$$n=\frac{4±14}{6}$$

Nā, me whakaoti te whārite n=\frac{4±14}{6} ina he tāpiri te ±. Tāpiri 4 ki te 14.

$$n=\frac{18}{6}$$

Whakawehe 18 ki te 6.

$$n=3$$

Nā, me whakaoti te whārite n=\frac{4±14}{6} ina he tango te ±. Tango 14 mai i 4.

$$n=-\frac{10}{6}$$

Whakahekea te hautanga \frac{-10}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Kua oti te whārite te whakatau.

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

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.

$$3n^{2}-4n-15=0$$

Me tāpiri 15 ki ngā taha e rua o te whārite.

$$3n^{2}-4n-15-\left(-15\right)=-\left(-15\right)$$

Mā te tango i te -15 i a ia ake anō ka toe ko te 0.

$$3n^{2}-4n=-\left(-15\right)$$

Tango -15 mai i 0.

$$3n^{2}-4n=15$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3n^{2}-4n}{3}=\frac{15}{3}$$

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

$$n^{2}-\frac{4}{3}n=\frac{15}{3}$$

Whakawehe 15 ki te 3.

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

Whakawehea te -\frac{4}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{2}{3}. Nā, tāpiria te pūrua o te -\frac{2}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}-\frac{4}{3}n+\left(-\frac{2}{3}\right)^{2}=5+\left(-\frac{2}{3}\right)^{2}$$

Pūruatia -\frac{2}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}-\frac{4}{3}n+\frac{4}{9}=5+\frac{4}{9}$$

Tāpiri 5 ki te \frac{4}{9}.

$$n^{2}-\frac{4}{3}n+\frac{4}{9}=\frac{49}{9}$$

Tauwehea n^{2}-\frac{4}{3}n+\frac{4}{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}.

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

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

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

Whakarūnātia.

$$n-\frac{2}{3}=\frac{7}{3} n-\frac{2}{3}=-\frac{7}{3}$$

Me tāpiri \frac{2}{3} ki ngā taha e rua o te whārite.

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