Whakaoti i te 25y^2-60y+36

$$25 y ^ { 2 } - 60 y + 36$$

Tauwehe

$$\left(5y-6\right)^{2}$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 25y^{2}+ay+by+36. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-60 ab=25\times 36=900$$

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 900.

$$-1,-900 -2,-450 -3,-300 -4,-225 -5,-180 -6,-150 -9,-100 -10,-90 -12,-75 -15,-60 -18,-50 -20,-45 -25,-36 -30,-30$$

Tātaihia te tapeke mō ia takirua.

$$-1-900=-901 -2-450=-452 -3-300=-303 -4-225=-229 -5-180=-185 -6-150=-156 -9-100=-109 -10-90=-100 -12-75=-87 -15-60=-75 -18-50=-68 -20-45=-65 -25-36=-61 -30-30=-60$$

Ko te otinga te takirua ka hoatu i te tapeke -60.

$$a=-30 b=-30$$

Tuhia anō te 25y^{2}-60y+36 hei \left(25y^{2}-30y\right)+\left(-30y+36\right).

$$\left(25y^{2}-30y\right)+\left(-30y+36\right)$$

Tauwehea te 5y i te tuatahi me te -6 i te rōpū tuarua.

$$5y\left(5y-6\right)-6\left(5y-6\right)$$

Whakatauwehea atu te kīanga pātahi 5y-6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(5y-6\right)\left(5y-6\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(5y-6\right)^{2}$$

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.

$$factor(25y^{2}-60y+36)$$

Kimihia te tauwehe pātahi nui rawa o ngā tau whakarea.

$$gcf(25,-60,36)=1$$

Kimihia te pūtakerua o te kīanga tau ārahi, 25y^{2}.

$$\sqrt{25y^{2}}=5y$$

Kimihia te pūtakerua o te kīanga tau autō, 36.

$$\sqrt{36}=6$$

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.

$$\left(5y-6\right)^{2}$$

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.

$$25y^{2}-60y+36=0$$

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.

$$y=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\times 25\times 36}}{2\times 25}$$

Pūrua -60.

$$y=\frac{-\left(-60\right)±\sqrt{3600-4\times 25\times 36}}{2\times 25}$$

Whakareatia -4 ki te 25.

$$y=\frac{-\left(-60\right)±\sqrt{3600-100\times 36}}{2\times 25}$$

Whakareatia -100 ki te 36.

$$y=\frac{-\left(-60\right)±\sqrt{3600-3600}}{2\times 25}$$

Tāpiri 3600 ki te -3600.

$$y=\frac{-\left(-60\right)±\sqrt{0}}{2\times 25}$$

Tuhia te pūtakerua o te 0.

$$y=\frac{-\left(-60\right)±0}{2\times 25}$$

Ko te tauaro o -60 ko 60.

$$y=\frac{60±0}{2\times 25}$$

Whakareatia 2 ki te 25.

$$y=\frac{60±0}{50}$$

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{6}{5} mō te x_{1} me te \frac{6}{5} mō te x_{2}.

$$25y^{2}-60y+36=25\left(y-\frac{6}{5}\right)\left(y-\frac{6}{5}\right)$$

Tango \frac{6}{5} 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.

$$25y^{2}-60y+36=25\times \frac{5y-6}{5}\left(y-\frac{6}{5}\right)$$

Tango \frac{6}{5} 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.

$$25y^{2}-60y+36=25\times \frac{5y-6}{5}\times \frac{5y-6}{5}$$

Whakareatia \frac{5y-6}{5} ki te \frac{5y-6}{5} 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.

$$25y^{2}-60y+36=25\times \frac{\left(5y-6\right)\left(5y-6\right)}{5\times 5}$$

Whakareatia 5 ki te 5.

$$25y^{2}-60y+36=25\times \frac{\left(5y-6\right)\left(5y-6\right)}{25}$$

Whakakorea atu te tauwehe pūnoa nui rawa 25 i roto i te 25 me te 25.

$$25y^{2}-60y+36=\left(5y-6\right)\left(5y-6\right)$$

Aromātai

$$\left(5y-6\right)^{2}$$