Whakaoti i te n^2-n-210=0

$$n ^ { 2 } - n - 210 = 0$$

Whakaoti mō n

$$n=-14$$

$$n=15$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te n^{2}-n-210 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.

$$a+b=-1 ab=-210$$

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 -210.

$$1,-210 2,-105 3,-70 5,-42 6,-35 7,-30 10,-21 14,-15$$

Tātaihia te tapeke mō ia takirua.

$$1-210=-209 2-105=-103 3-70=-67 5-42=-37 6-35=-29 7-30=-23 10-21=-11 14-15=-1$$

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

$$a=-15 b=14$$

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

$$\left(n-15\right)\left(n+14\right)$$

Hei kimi otinga whārite, me whakaoti te n-15=0 me te n+14=0.

$$n=15 n=-14$$

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 n^{2}+an+bn-210. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=1\left(-210\right)=-210$$

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 -210.

$$1,-210 2,-105 3,-70 5,-42 6,-35 7,-30 10,-21 14,-15$$

Tātaihia te tapeke mō ia takirua.

$$1-210=-209 2-105=-103 3-70=-67 5-42=-37 6-35=-29 7-30=-23 10-21=-11 14-15=-1$$

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

$$a=-15 b=14$$

Tuhia anō te n^{2}-n-210 hei \left(n^{2}-15n\right)+\left(14n-210\right).

$$\left(n^{2}-15n\right)+\left(14n-210\right)$$

Tauwehea te n i te tuatahi me te 14 i te rōpū tuarua.

$$n\left(n-15\right)+14\left(n-15\right)$$

Whakatauwehea atu te kīanga pātahi n-15 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(n-15\right)\left(n+14\right)$$

Hei kimi otinga whārite, me whakaoti te n-15=0 me te n+14=0.

$$n=15 n=-14$$

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.

$$n^{2}-n-210=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -1 mō b, me -210 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-1\right)±\sqrt{1-4\left(-210\right)}}{2}$$

Whakareatia -4 ki te -210.

$$n=\frac{-\left(-1\right)±\sqrt{1+840}}{2}$$

Tāpiri 1 ki te 840.

$$n=\frac{-\left(-1\right)±\sqrt{841}}{2}$$

Tuhia te pūtakerua o te 841.

$$n=\frac{-\left(-1\right)±29}{2}$$

Ko te tauaro o -1 ko 1.

$$n=\frac{1±29}{2}$$

Nā, me whakaoti te whārite n=\frac{1±29}{2} ina he tāpiri te ±. Tāpiri 1 ki te 29.

$$n=\frac{30}{2}$$

Whakawehe 30 ki te 2.

$$n=15$$

Nā, me whakaoti te whārite n=\frac{1±29}{2} ina he tango te ±. Tango 29 mai i 1.

$$n=-\frac{28}{2}$$

Whakawehe -28 ki te 2.

$$n=-14$$

Kua oti te whārite te whakatau.

$$n=15 n=-14$$

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.

$$n^{2}-n-210=0$$

Me tāpiri 210 ki ngā taha e rua o te whārite.

$$n^{2}-n-210-\left(-210\right)=-\left(-210\right)$$

Mā te tango i te -210 i a ia ake anō ka toe ko te 0.

$$n^{2}-n=-\left(-210\right)$$

Tango -210 mai i 0.

$$n^{2}-n=210$$

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.

$$n^{2}-n+\left(-\frac{1}{2}\right)^{2}=210+\left(-\frac{1}{2}\right)^{2}$$

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

$$n^{2}-n+\frac{1}{4}=210+\frac{1}{4}$$

Tāpiri 210 ki te \frac{1}{4}.

$$n^{2}-n+\frac{1}{4}=\frac{841}{4}$$

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

$$\left(n-\frac{1}{2}\right)^{2}=\frac{841}{4}$$

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

$$\sqrt{\left(n-\frac{1}{2}\right)^{2}}=\sqrt{\frac{841}{4}}$$

Whakarūnātia.

$$n-\frac{1}{2}=\frac{29}{2} n-\frac{1}{2}=-\frac{29}{2}$$

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

$$n=15 n=-14$$