Whakaoti i te x^2-18x+81

$$x ^ { 2 } - 18 x + 81$$

Tauwehe

$$\left(x-9\right)^{2}$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei x^{2}+ax+bx+81. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

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

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro 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 x^{2}-18x+81 hei \left(x^{2}-9x\right)+\left(-9x+81\right).

$$\left(x^{2}-9x\right)+\left(-9x+81\right)$$

Tauwehea te x i te tuatahi me te -9 i te rōpū tuarua.

$$x\left(x-9\right)-9\left(x-9\right)$$

Whakatauwehea atu te kīanga pātahi x-9 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-9\right)\left(x-9\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(x-9\right)^{2}$$

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.

$$factor(x^{2}-18x+81)$$

Kimihia te pūtakerua o te kīanga tau autō, 81.

$$\sqrt{81}=9$$

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.

$$\left(x-9\right)^{2}$$

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.

$$x^{2}-18x+81=0$$

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.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 81}}{2}$$

Pūrua -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 81}}{2}$$

Whakareatia -4 ki te 81.

$$x=\frac{-\left(-18\right)±\sqrt{324-324}}{2}$$

Tāpiri 324 ki te -324.

$$x=\frac{-\left(-18\right)±\sqrt{0}}{2}$$

Tuhia te pūtakerua o te 0.

$$x=\frac{-\left(-18\right)±0}{2}$$

Ko te tauaro o -18 ko 18.

$$x=\frac{18±0}{2}$$

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 9 mō te x_{1} me te 9 mō te x_{2}.

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

Aromātai

$$\left(x-9\right)^{2}$$