Whakaoti i te 81n^2+18n+1

$$81 n ^ { 2 } + 18 n + 1$$

Tauwehe

$$\left(9n+1\right)^{2}$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 81n^{2}+an+bn+1. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=18 ab=81\times 1=81$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 81.

$$1,81 3,27 9,9$$

Tātaihia te tapeke mō ia takirua.

$$1+81=82 3+27=30 9+9=18$$

Ko te otinga te takirua ka hoatu i te tapeke 18.

$$a=9 b=9$$

Tuhia anō te 81n^{2}+18n+1 hei \left(81n^{2}+9n\right)+\left(9n+1\right).

$$\left(81n^{2}+9n\right)+\left(9n+1\right)$$

Whakatauwehea atu 9n i te 81n^{2}+9n.

$$9n\left(9n+1\right)+9n+1$$

Whakatauwehea atu te kīanga pātahi 9n+1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(9n+1\right)\left(9n+1\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(9n+1\right)^{2}$$

Ngā Upane Whakamahi Tauwehe o te Pūrua Huarua

Ko te tikanga tātai o tēnei huatoru he pūrua huatoru, ka whakareatia pea e tētahi tauwehe pātahi. Ka taea ngā pūrua huatoru te tauwehe mā te kimi i ngā pūtakerua o ngā kīanga tau ārahi, autō hoki.

$$factor(81n^{2}+18n+1)$$

Kimihia te tauwehe pātahi nui rawa o ngā tau whakarea.

$$gcf(81,18,1)=1$$

Kimihia te pūtakerua o te kīanga tau ārahi, 81n^{2}.

$$\sqrt{81n^{2}}=9n$$

Ko te pūrua huatoru te pūrua o te huarua ko te tapeke tērā, te huatango rānei o ngā pūtakerua o ngā kīanga tau ārahi, autō hoki, e whakaritea ai te tohu e te tohu o te kīanga tau waenga o te pūrua huatoru.

$$\left(9n+1\right)^{2}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Ka taea te huamaha pūrua te tauwehe mā te whakamahi i te huringa ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), ina ko x_{1} me x_{2} ngā otinga o te whārite pūrua ax^{2}+bx+c=0.

$$81n^{2}+18n+1=0$$

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.

$$n=\frac{-18±\sqrt{18^{2}-4\times 81}}{2\times 81}$$

Pūrua 18.

$$n=\frac{-18±\sqrt{324-4\times 81}}{2\times 81}$$

Whakareatia -4 ki te 81.

$$n=\frac{-18±\sqrt{324-324}}{2\times 81}$$

Tāpiri 324 ki te -324.

$$n=\frac{-18±\sqrt{0}}{2\times 81}$$

Tuhia te pūtakerua o te 0.

$$n=\frac{-18±0}{2\times 81}$$

Whakareatia 2 ki te 81.

$$n=\frac{-18±0}{162}$$

Tauwehea te kīanga taketake mā te whakamahi i te ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Me whakakapi te -\frac{1}{9} mō te x_{1} me te -\frac{1}{9} mō te x_{2}.

$$81n^{2}+18n+1=81\left(n-\left(-\frac{1}{9}\right)\right)\left(n-\left(-\frac{1}{9}\right)\right)$$

Whakamāmātia ngā kīanga katoa o te āhua p-\left(-q\right) ki te p+q.

$$81n^{2}+18n+1=81\left(n+\frac{1}{9}\right)\left(n+\frac{1}{9}\right)$$

Tāpiri \frac{1}{9} ki te n 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.

$$81n^{2}+18n+1=81\times \frac{9n+1}{9}\left(n+\frac{1}{9}\right)$$

Tāpiri \frac{1}{9} ki te n 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.

$$81n^{2}+18n+1=81\times \frac{9n+1}{9}\times \frac{9n+1}{9}$$

Whakareatia \frac{9n+1}{9} ki te \frac{9n+1}{9} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$81n^{2}+18n+1=81\times \frac{\left(9n+1\right)\left(9n+1\right)}{9\times 9}$$

Whakareatia 9 ki te 9.

$$81n^{2}+18n+1=81\times \frac{\left(9n+1\right)\left(9n+1\right)}{81}$$

Whakakorea atu te tauwehe pūnoa nui rawa 81 i roto i te 81 me te 81.

$$81n^{2}+18n+1=\left(9n+1\right)\left(9n+1\right)$$

Aromātai

$$\left(9n+1\right)^{2}$$