Whakaoti i te 561=2n^2-n

$$561 = 2 n ^ { 2 } - n$$

Whakaoti mō n

$$n = -\frac{33}{2} = -16\frac{1}{2} = -16.5$$

$$n=17$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

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

$$2n^{2}-n=561$$

Tangohia te 561 mai i ngā taha e rua.

$$2n^{2}-n-561=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 2n^{2}+an+bn-561. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=2\left(-561\right)=-1122$$

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 -1122.

$$1,-1122 2,-561 3,-374 6,-187 11,-102 17,-66 22,-51 33,-34$$

Tātaihia te tapeke mō ia takirua.

$$1-1122=-1121 2-561=-559 3-374=-371 6-187=-181 11-102=-91 17-66=-49 22-51=-29 33-34=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -1.

$$a=-34 b=33$$

Tuhia anō te 2n^{2}-n-561 hei \left(2n^{2}-34n\right)+\left(33n-561\right).

$$\left(2n^{2}-34n\right)+\left(33n-561\right)$$

Tauwehea te 2n i te tuatahi me te 33 i te rōpū tuarua.

$$2n\left(n-17\right)+33\left(n-17\right)$$

Whakatauwehea atu te kīanga pātahi n-17 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(n-17\right)\left(2n+33\right)$$

Hei kimi otinga whārite, me whakaoti te n-17=0 me te 2n+33=0.

$$n=17 n=-\frac{33}{2}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$2n^{2}-n=561$$

Tangohia te 561 mai i ngā taha e rua.

$$2n^{2}-n-561=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -1 mō b, me -561 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-1\right)±\sqrt{1-4\times 2\left(-561\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$n=\frac{-\left(-1\right)±\sqrt{1-8\left(-561\right)}}{2\times 2}$$

Whakareatia -8 ki te -561.

$$n=\frac{-\left(-1\right)±\sqrt{1+4488}}{2\times 2}$$

Tāpiri 1 ki te 4488.

$$n=\frac{-\left(-1\right)±\sqrt{4489}}{2\times 2}$$

Tuhia te pūtakerua o te 4489.

$$n=\frac{-\left(-1\right)±67}{2\times 2}$$

Ko te tauaro o -1 ko 1.

$$n=\frac{1±67}{2\times 2}$$

Whakareatia 2 ki te 2.

$$n=\frac{1±67}{4}$$

Nā, me whakaoti te whārite n=\frac{1±67}{4} ina he tāpiri te ±. Tāpiri 1 ki te 67.

$$n=\frac{68}{4}$$

Whakawehe 68 ki te 4.

$$n=17$$

Nā, me whakaoti te whārite n=\frac{1±67}{4} ina he tango te ±. Tango 67 mai i 1.

$$n=-\frac{66}{4}$$

Whakahekea te hautanga \frac{-66}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$n=-\frac{33}{2}$$

Kua oti te whārite te whakatau.

$$n=17 n=-\frac{33}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$2n^{2}-n=561$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2n^{2}-n}{2}=\frac{561}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$n^{2}-\frac{1}{2}n=\frac{561}{2}$$

Whakawehea te -\frac{1}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{4}. Nā, tāpiria te pūrua o te -\frac{1}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}-\frac{1}{2}n+\left(-\frac{1}{4}\right)^{2}=\frac{561}{2}+\left(-\frac{1}{4}\right)^{2}$$

Pūruatia -\frac{1}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}-\frac{1}{2}n+\frac{1}{16}=\frac{561}{2}+\frac{1}{16}$$

Tāpiri \frac{561}{2} ki te \frac{1}{16} 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.

$$n^{2}-\frac{1}{2}n+\frac{1}{16}=\frac{4489}{16}$$

Tauwehea n^{2}-\frac{1}{2}n+\frac{1}{16}. 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}.

$$\left(n-\frac{1}{4}\right)^{2}=\frac{4489}{16}$$

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

$$\sqrt{\left(n-\frac{1}{4}\right)^{2}}=\sqrt{\frac{4489}{16}}$$

Whakarūnātia.

$$n-\frac{1}{4}=\frac{67}{4} n-\frac{1}{4}=-\frac{67}{4}$$

Me tāpiri \frac{1}{4} ki ngā taha e rua o te whārite.

$$n=17 n=-\frac{33}{2}$$