Whakaoti i te 3n^2-11n-874=0

$$3 { n }^{ 2 } -11n-874 = 0$$

Whakaoti mō n

$$n = -\frac{46}{3} = -15\frac{1}{3} \approx -15.333333333$$

$$n=19$$

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-874. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-11 ab=3\left(-874\right)=-2622$$

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 -2622.

$$1,-2622 2,-1311 3,-874 6,-437 19,-138 23,-114 38,-69 46,-57$$

Tātaihia te tapeke mō ia takirua.

$$1-2622=-2621 2-1311=-1309 3-874=-871 6-437=-431 19-138=-119 23-114=-91 38-69=-31 46-57=-11$$

Ko te otinga te takirua ka hoatu i te tapeke -11.

$$a=-57 b=46$$

Tuhia anō te 3n^{2}-11n-874 hei \left(3n^{2}-57n\right)+\left(46n-874\right).

$$\left(3n^{2}-57n\right)+\left(46n-874\right)$$

Tauwehea te 3n i te tuatahi me te 46 i te rōpū tuarua.

$$3n\left(n-19\right)+46\left(n-19\right)$$

Whakatauwehea atu te kīanga pātahi n-19 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(n-19\right)\left(3n+46\right)$$

Hei kimi otinga whārite, me whakaoti te n-19=0 me te 3n+46=0.

$$n=19 n=-\frac{46}{3}$$

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.

$$3n^{2}-11n-874=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -11 mō b, me -874 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 3\left(-874\right)}}{2\times 3}$$

Pūrua -11.

$$n=\frac{-\left(-11\right)±\sqrt{121-4\times 3\left(-874\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$n=\frac{-\left(-11\right)±\sqrt{121-12\left(-874\right)}}{2\times 3}$$

Whakareatia -12 ki te -874.

$$n=\frac{-\left(-11\right)±\sqrt{121+10488}}{2\times 3}$$

Tāpiri 121 ki te 10488.

$$n=\frac{-\left(-11\right)±\sqrt{10609}}{2\times 3}$$

Tuhia te pūtakerua o te 10609.

$$n=\frac{-\left(-11\right)±103}{2\times 3}$$

Ko te tauaro o -11 ko 11.

$$n=\frac{11±103}{2\times 3}$$

Whakareatia 2 ki te 3.

$$n=\frac{11±103}{6}$$

Nā, me whakaoti te whārite n=\frac{11±103}{6} ina he tāpiri te ±. Tāpiri 11 ki te 103.

$$n=\frac{114}{6}$$

Whakawehe 114 ki te 6.

$$n=19$$

Nā, me whakaoti te whārite n=\frac{11±103}{6} ina he tango te ±. Tango 103 mai i 11.

$$n=-\frac{92}{6}$$

Whakahekea te hautanga \frac{-92}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$n=-\frac{46}{3}$$

Kua oti te whārite te whakatau.

$$n=19 n=-\frac{46}{3}$$

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.

$$3n^{2}-11n-874=0$$

Me tāpiri 874 ki ngā taha e rua o te whārite.

$$3n^{2}-11n-874-\left(-874\right)=-\left(-874\right)$$

Mā te tango i te -874 i a ia ake anō ka toe ko te 0.

$$3n^{2}-11n=-\left(-874\right)$$

Tango -874 mai i 0.

$$3n^{2}-11n=874$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3n^{2}-11n}{3}=\frac{874}{3}$$

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

$$n^{2}-\frac{11}{3}n=\frac{874}{3}$$

Whakawehea te -\frac{11}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{6}. Nā, tāpiria te pūrua o te -\frac{11}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}-\frac{11}{3}n+\left(-\frac{11}{6}\right)^{2}=\frac{874}{3}+\left(-\frac{11}{6}\right)^{2}$$

Pūruatia -\frac{11}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}-\frac{11}{3}n+\frac{121}{36}=\frac{874}{3}+\frac{121}{36}$$

Tāpiri \frac{874}{3} ki te \frac{121}{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.

$$n^{2}-\frac{11}{3}n+\frac{121}{36}=\frac{10609}{36}$$

Tauwehea n^{2}-\frac{11}{3}n+\frac{121}{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}.

$$\left(n-\frac{11}{6}\right)^{2}=\frac{10609}{36}$$

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

$$\sqrt{\left(n-\frac{11}{6}\right)^{2}}=\sqrt{\frac{10609}{36}}$$

Whakarūnātia.

$$n-\frac{11}{6}=\frac{103}{6} n-\frac{11}{6}=-\frac{103}{6}$$

Me tāpiri \frac{11}{6} ki ngā taha e rua o te whārite.

$$n=19 n=-\frac{46}{3}$$