Whakaoti i te 636=5n+4n^2

$$636 = 5 n + 4 n ^ { 2 }$$

Whakaoti mō n

$$n = -\frac{53}{4} = -13\frac{1}{4} = -13.25$$

$$n=12$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$5n+4n^{2}=636$$

Tangohia te 636 mai i ngā taha e rua.

$$5n+4n^{2}-636=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$4n^{2}+5n-636=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 4n^{2}+an+bn-636. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=5 ab=4\left(-636\right)=-2544$$

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 -2544.

$$-1,2544 -2,1272 -3,848 -4,636 -6,424 -8,318 -12,212 -16,159 -24,106 -48,53$$

Tātaihia te tapeke mō ia takirua.

$$-1+2544=2543 -2+1272=1270 -3+848=845 -4+636=632 -6+424=418 -8+318=310 -12+212=200 -16+159=143 -24+106=82 -48+53=5$$

Ko te otinga te takirua ka hoatu i te tapeke 5.

$$a=-48 b=53$$

Tuhia anō te 4n^{2}+5n-636 hei \left(4n^{2}-48n\right)+\left(53n-636\right).

$$\left(4n^{2}-48n\right)+\left(53n-636\right)$$

Tauwehea te 4n i te tuatahi me te 53 i te rōpū tuarua.

$$4n\left(n-12\right)+53\left(n-12\right)$$

Whakatauwehea atu te kīanga pātahi n-12 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(n-12\right)\left(4n+53\right)$$

Hei kimi otinga whārite, me whakaoti te n-12=0 me te 4n+53=0.

$$n=12 n=-\frac{53}{4}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$5n+4n^{2}=636$$

Tangohia te 636 mai i ngā taha e rua.

$$5n+4n^{2}-636=0$$

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.

$$4n^{2}+5n-636=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 5 mō b, me -636 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-5±\sqrt{5^{2}-4\times 4\left(-636\right)}}{2\times 4}$$

Pūrua 5.

$$n=\frac{-5±\sqrt{25-4\times 4\left(-636\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$n=\frac{-5±\sqrt{25-16\left(-636\right)}}{2\times 4}$$

Whakareatia -16 ki te -636.

$$n=\frac{-5±\sqrt{25+10176}}{2\times 4}$$

Tāpiri 25 ki te 10176.

$$n=\frac{-5±\sqrt{10201}}{2\times 4}$$

Tuhia te pūtakerua o te 10201.

$$n=\frac{-5±101}{2\times 4}$$

Whakareatia 2 ki te 4.

$$n=\frac{-5±101}{8}$$

Nā, me whakaoti te whārite n=\frac{-5±101}{8} ina he tāpiri te ±. Tāpiri -5 ki te 101.

$$n=\frac{96}{8}$$

Whakawehe 96 ki te 8.

$$n=12$$

Nā, me whakaoti te whārite n=\frac{-5±101}{8} ina he tango te ±. Tango 101 mai i -5.

$$n=-\frac{106}{8}$$

Whakahekea te hautanga \frac{-106}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$n=-\frac{53}{4}$$

Kua oti te whārite te whakatau.

$$n=12 n=-\frac{53}{4}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$5n+4n^{2}=636$$

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.

$$4n^{2}+5n=636$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4n^{2}+5n}{4}=\frac{636}{4}$$

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

$$n^{2}+\frac{5}{4}n=\frac{636}{4}$$

Whakawehe 636 ki te 4.

$$n^{2}+\frac{5}{4}n=159$$

Whakawehea te \frac{5}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{8}. Nā, tāpiria te pūrua o te \frac{5}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}+\frac{5}{4}n+\left(\frac{5}{8}\right)^{2}=159+\left(\frac{5}{8}\right)^{2}$$

Pūruatia \frac{5}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}+\frac{5}{4}n+\frac{25}{64}=159+\frac{25}{64}$$

Tāpiri 159 ki te \frac{25}{64}.

$$n^{2}+\frac{5}{4}n+\frac{25}{64}=\frac{10201}{64}$$

Tauwehea te n^{2}+\frac{5}{4}n+\frac{25}{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}.

$$\left(n+\frac{5}{8}\right)^{2}=\frac{10201}{64}$$

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

$$\sqrt{\left(n+\frac{5}{8}\right)^{2}}=\sqrt{\frac{10201}{64}}$$

Whakarūnātia.

$$n+\frac{5}{8}=\frac{101}{8} n+\frac{5}{8}=-\frac{101}{8}$$

Me tango \frac{5}{8} mai i ngā taha e rua o te whārite.

$$n=12 n=-\frac{53}{4}$$