Whakaoti i te n^2-n=272

$$n ^ { 2 } - n = 272$$

Whakaoti mō n

$$n=-16$$

$$n=17$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 272 mai i ngā taha e rua.

$$n^{2}-n-272=0$$

Hei whakaoti i te whārite, whakatauwehea te n^{2}-n-272 mā te whakamahi i te tātai n^{2}+\left(a+b\right)n+ab=\left(n+a\right)\left(n+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=-272$$

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 -272.

$$1,-272 2,-136 4,-68 8,-34 16,-17$$

Tātaihia te tapeke mō ia takirua.

$$1-272=-271 2-136=-134 4-68=-64 8-34=-26 16-17=-1$$

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

$$a=-17 b=16$$

Me tuhi anō te kīanga whakatauwehe \left(n+a\right)\left(n+b\right) mā ngā uara i tātaihia.

$$\left(n-17\right)\left(n+16\right)$$

Hei kimi otinga whārite, me whakaoti te n-17=0 me te n+16=0.

$$n=17 n=-16$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 272 mai i ngā taha e rua.

$$n^{2}-n-272=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei n^{2}+an+bn-272. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=1\left(-272\right)=-272$$

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 -272.

$$1,-272 2,-136 4,-68 8,-34 16,-17$$

Tātaihia te tapeke mō ia takirua.

$$1-272=-271 2-136=-134 4-68=-64 8-34=-26 16-17=-1$$

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

$$a=-17 b=16$$

Tuhia anō te n^{2}-n-272 hei \left(n^{2}-17n\right)+\left(16n-272\right).

$$\left(n^{2}-17n\right)+\left(16n-272\right)$$

Tauwehea te n i te tuatahi me te 16 i te rōpū tuarua.

$$n\left(n-17\right)+16\left(n-17\right)$$

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

$$\left(n-17\right)\left(n+16\right)$$

Hei kimi otinga whārite, me whakaoti te n-17=0 me te n+16=0.

$$n=17 n=-16$$

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.

$$n^{2}-n=272$$

Me tango 272 mai i ngā taha e rua o te whārite.

$$n^{2}-n-272=272-272$$

Mā te tango i te 272 i a ia ake anō ka toe ko te 0.

$$n^{2}-n-272=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -1 mō b, me -272 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-1\right)±\sqrt{1-4\left(-272\right)}}{2}$$

Whakareatia -4 ki te -272.

$$n=\frac{-\left(-1\right)±\sqrt{1+1088}}{2}$$

Tāpiri 1 ki te 1088.

$$n=\frac{-\left(-1\right)±\sqrt{1089}}{2}$$

Tuhia te pūtakerua o te 1089.

$$n=\frac{-\left(-1\right)±33}{2}$$

Ko te tauaro o -1 ko 1.

$$n=\frac{1±33}{2}$$

Nā, me whakaoti te whārite n=\frac{1±33}{2} ina he tāpiri te ±. Tāpiri 1 ki te 33.

$$n=\frac{34}{2}$$

Whakawehe 34 ki te 2.

$$n=17$$

Nā, me whakaoti te whārite n=\frac{1±33}{2} ina he tango te ±. Tango 33 mai i 1.

$$n=-\frac{32}{2}$$

Whakawehe -32 ki te 2.

$$n=-16$$

Kua oti te whārite te whakatau.

$$n=17 n=-16$$

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.

$$n^{2}-n=272$$

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

$$n^{2}-n+\left(-\frac{1}{2}\right)^{2}=272+\left(-\frac{1}{2}\right)^{2}$$

Pūruatia -\frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}-n+\frac{1}{4}=272+\frac{1}{4}$$

Tāpiri 272 ki te \frac{1}{4}.

$$n^{2}-n+\frac{1}{4}=\frac{1089}{4}$$

Tauwehea n^{2}-n+\frac{1}{4}. 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}.

$$\left(n-\frac{1}{2}\right)^{2}=\frac{1089}{4}$$

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

$$\sqrt{\left(n-\frac{1}{2}\right)^{2}}=\sqrt{\frac{1089}{4}}$$

Whakarūnātia.

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

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

$$n=17 n=-16$$