Whakaoti i te 4n^2-7n=11

$$4 n ^ { 2 } - 7 n = 11$$

Whakaoti mō n

$$n=-1$$

$$n = \frac{11}{4} = 2\frac{3}{4} = 2.75$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 11 mai i ngā taha e rua.

$$4n^{2}-7n-11=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 4n^{2}+an+bn-11. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-7 ab=4\left(-11\right)=-44$$

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 -44.

$$1,-44 2,-22 4,-11$$

Tātaihia te tapeke mō ia takirua.

$$1-44=-43 2-22=-20 4-11=-7$$

Ko te otinga te takirua ka hoatu i te tapeke -7.

$$a=-11 b=4$$

Tuhia anō te 4n^{2}-7n-11 hei \left(4n^{2}-11n\right)+\left(4n-11\right).

$$\left(4n^{2}-11n\right)+\left(4n-11\right)$$

Whakatauwehea atu n i te 4n^{2}-11n.

$$n\left(4n-11\right)+4n-11$$

Whakatauwehea atu te kīanga pātahi 4n-11 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(4n-11\right)\left(n+1\right)$$

Hei kimi otinga whārite, me whakaoti te 4n-11=0 me te n+1=0.

$$n=\frac{11}{4} n=-1$$

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.

$$4n^{2}-7n=11$$

Me tango 11 mai i ngā taha e rua o te whārite.

$$4n^{2}-7n-11=11-11$$

Mā te tango i te 11 i a ia ake anō ka toe ko te 0.

$$4n^{2}-7n-11=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -7 mō b, me -11 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 4\left(-11\right)}}{2\times 4}$$

Pūrua -7.

$$n=\frac{-\left(-7\right)±\sqrt{49-4\times 4\left(-11\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$n=\frac{-\left(-7\right)±\sqrt{49-16\left(-11\right)}}{2\times 4}$$

Whakareatia -16 ki te -11.

$$n=\frac{-\left(-7\right)±\sqrt{49+176}}{2\times 4}$$

Tāpiri 49 ki te 176.

$$n=\frac{-\left(-7\right)±\sqrt{225}}{2\times 4}$$

Tuhia te pūtakerua o te 225.

$$n=\frac{-\left(-7\right)±15}{2\times 4}$$

Ko te tauaro o -7 ko 7.

$$n=\frac{7±15}{2\times 4}$$

Whakareatia 2 ki te 4.

$$n=\frac{7±15}{8}$$

Nā, me whakaoti te whārite n=\frac{7±15}{8} ina he tāpiri te ±. Tāpiri 7 ki te 15.

$$n=\frac{22}{8}$$

Whakahekea te hautanga \frac{22}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$n=\frac{11}{4}$$

Nā, me whakaoti te whārite n=\frac{7±15}{8} ina he tango te ±. Tango 15 mai i 7.

$$n=-\frac{8}{8}$$

Whakawehe -8 ki te 8.

$$n=-1$$

Kua oti te whārite te whakatau.

$$n=\frac{11}{4} n=-1$$

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.

$$4n^{2}-7n=11$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4n^{2}-7n}{4}=\frac{11}{4}$$

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

$$n^{2}-\frac{7}{4}n=\frac{11}{4}$$

Whakawehea te -\frac{7}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{8}. Nā, tāpiria te pūrua o te -\frac{7}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}-\frac{7}{4}n+\left(-\frac{7}{8}\right)^{2}=\frac{11}{4}+\left(-\frac{7}{8}\right)^{2}$$

Pūruatia -\frac{7}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}-\frac{7}{4}n+\frac{49}{64}=\frac{11}{4}+\frac{49}{64}$$

Tāpiri \frac{11}{4} ki te \frac{49}{64} 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.

$$n^{2}-\frac{7}{4}n+\frac{49}{64}=\frac{225}{64}$$

Tauwehea te n^{2}-\frac{7}{4}n+\frac{49}{64}. 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}.

$$\left(n-\frac{7}{8}\right)^{2}=\frac{225}{64}$$

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

$$\sqrt{\left(n-\frac{7}{8}\right)^{2}}=\sqrt{\frac{225}{64}}$$

Whakarūnātia.

$$n-\frac{7}{8}=\frac{15}{8} n-\frac{7}{8}=-\frac{15}{8}$$

Me tāpiri \frac{7}{8} ki ngā taha e rua o te whārite.

$$n=\frac{11}{4} n=-1$$