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

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

Tauwehe

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

Ngā Upane Otinga

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

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

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

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

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

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

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

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

Tuhia anōtia hei pūrua huarua.

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

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.

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

Kimihia te tauwehe pātahi nui rawa o ngā tau whakarea.

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

Kimihia te pūtakerua o te kīanga tau ārahi, 4y^{2}.

$$\sqrt{4y^{2}}=2y$$

Kimihia te pūtakerua o te kīanga tau autō, 9.

$$\sqrt{9}=3$$

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.

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

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.

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

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.

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

Pūrua -12.

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

Whakareatia -4 ki te 4.

$$y=\frac{-\left(-12\right)±\sqrt{144-16\times 9}}{2\times 4}$$

Whakareatia -16 ki te 9.

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

Tāpiri 144 ki te -144.

$$y=\frac{-\left(-12\right)±\sqrt{0}}{2\times 4}$$

Tuhia te pūtakerua o te 0.

$$y=\frac{-\left(-12\right)±0}{2\times 4}$$

Ko te tauaro o -12 ko 12.

$$y=\frac{12±0}{2\times 4}$$

Whakareatia 2 ki te 4.

$$y=\frac{12±0}{8}$$

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

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

Tango \frac{3}{2} 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.

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

Tango \frac{3}{2} 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.

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

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

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

Whakareatia 2 ki te 2.

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

Whakakorea atu te tauwehe pūnoa nui rawa 4 i roto i te 4 me te 4.

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

Aromātai

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