Whakaoti i te 9y^2-12y+4

$$9 y ^ { 2 } - 12 y + 4$$

Tauwehe

$$\left(3y-2\right)^{2}$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 9y^{2}+ay+by+4. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-12 ab=9\times 4=36$$

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 36.

$$-1,-36 -2,-18 -3,-12 -4,-9 -6,-6$$

Tātaihia te tapeke mō ia takirua.

$$-1-36=-37 -2-18=-20 -3-12=-15 -4-9=-13 -6-6=-12$$

Ko te otinga te takirua ka hoatu i te tapeke -12.

$$a=-6 b=-6$$

Tuhia anō te 9y^{2}-12y+4 hei \left(9y^{2}-6y\right)+\left(-6y+4\right).

$$\left(9y^{2}-6y\right)+\left(-6y+4\right)$$

Tauwehea te 3y i te tuatahi me te -2 i te rōpū tuarua.

$$3y\left(3y-2\right)-2\left(3y-2\right)$$

Whakatauwehea atu te kīanga pātahi 3y-2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3y-2\right)\left(3y-2\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(3y-2\right)^{2}$$

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.

$$factor(9y^{2}-12y+4)$$

Kimihia te tauwehe pātahi nui rawa o ngā tau whakarea.

$$gcf(9,-12,4)=1$$

Kimihia te pūtakerua o te kīanga tau ārahi, 9y^{2}.

$$\sqrt{9y^{2}}=3y$$

Kimihia te pūtakerua o te kīanga tau autō, 4.

$$\sqrt{4}=2$$

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.

$$\left(3y-2\right)^{2}$$

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.

$$9y^{2}-12y+4=0$$

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.

$$y=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 9\times 4}}{2\times 9}$$

Pūrua -12.

$$y=\frac{-\left(-12\right)±\sqrt{144-4\times 9\times 4}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$y=\frac{-\left(-12\right)±\sqrt{144-36\times 4}}{2\times 9}$$

Whakareatia -36 ki te 4.

$$y=\frac{-\left(-12\right)±\sqrt{144-144}}{2\times 9}$$

Tāpiri 144 ki te -144.

$$y=\frac{-\left(-12\right)±\sqrt{0}}{2\times 9}$$

Tuhia te pūtakerua o te 0.

$$y=\frac{-\left(-12\right)±0}{2\times 9}$$

Ko te tauaro o -12 ko 12.

$$y=\frac{12±0}{2\times 9}$$

Whakareatia 2 ki te 9.

$$y=\frac{12±0}{18}$$

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{2}{3} mō te x_{1} me te \frac{2}{3} mō te x_{2}.

$$9y^{2}-12y+4=9\left(y-\frac{2}{3}\right)\left(y-\frac{2}{3}\right)$$

Tango \frac{2}{3} mai i y mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$9y^{2}-12y+4=9\times \frac{3y-2}{3}\left(y-\frac{2}{3}\right)$$

Tango \frac{2}{3} mai i y mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$9y^{2}-12y+4=9\times \frac{3y-2}{3}\times \frac{3y-2}{3}$$

Whakareatia \frac{3y-2}{3} ki te \frac{3y-2}{3} 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.

$$9y^{2}-12y+4=9\times \frac{\left(3y-2\right)\left(3y-2\right)}{3\times 3}$$

Whakareatia 3 ki te 3.

$$9y^{2}-12y+4=9\times \frac{\left(3y-2\right)\left(3y-2\right)}{9}$$

Whakakorea atu te tauwehe pūnoa nui rawa 9 i roto i te 9 me te 9.

$$9y^{2}-12y+4=\left(3y-2\right)\left(3y-2\right)$$

Aromātai

$$\left(3y-2\right)^{2}$$