Whakaoti i te 25n^2-30n+9

$$25 n ^ { 2 } - 30 n + 9$$

Tauwehe

$$\left(5n-3\right)^{2}$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 25n^{2}+an+bn+9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-30 ab=25\times 9=225$$

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 225.

$$-1,-225 -3,-75 -5,-45 -9,-25 -15,-15$$

Tātaihia te tapeke mō ia takirua.

$$-1-225=-226 -3-75=-78 -5-45=-50 -9-25=-34 -15-15=-30$$

Ko te otinga te takirua ka hoatu i te tapeke -30.

$$a=-15 b=-15$$

Tuhia anō te 25n^{2}-30n+9 hei \left(25n^{2}-15n\right)+\left(-15n+9\right).

$$\left(25n^{2}-15n\right)+\left(-15n+9\right)$$

Tauwehea te 5n i te tuatahi me te -3 i te rōpū tuarua.

$$5n\left(5n-3\right)-3\left(5n-3\right)$$

Whakatauwehea atu te kīanga pātahi 5n-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(5n-3\right)\left(5n-3\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(5n-3\right)^{2}$$

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.

$$factor(25n^{2}-30n+9)$$

Kimihia te tauwehe pātahi nui rawa o ngā tau whakarea.

$$gcf(25,-30,9)=1$$

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

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

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.

$$\left(5n-3\right)^{2}$$

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.

$$25n^{2}-30n+9=0$$

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.

$$n=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 25\times 9}}{2\times 25}$$

Pūrua -30.

$$n=\frac{-\left(-30\right)±\sqrt{900-4\times 25\times 9}}{2\times 25}$$

Whakareatia -4 ki te 25.

$$n=\frac{-\left(-30\right)±\sqrt{900-100\times 9}}{2\times 25}$$

Whakareatia -100 ki te 9.

$$n=\frac{-\left(-30\right)±\sqrt{900-900}}{2\times 25}$$

Tāpiri 900 ki te -900.

$$n=\frac{-\left(-30\right)±\sqrt{0}}{2\times 25}$$

Tuhia te pūtakerua o te 0.

$$n=\frac{-\left(-30\right)±0}{2\times 25}$$

Ko te tauaro o -30 ko 30.

$$n=\frac{30±0}{2\times 25}$$

Whakareatia 2 ki te 25.

$$n=\frac{30±0}{50}$$

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{3}{5} mō te x_{1} me te \frac{3}{5} mō te x_{2}.

$$25n^{2}-30n+9=25\left(n-\frac{3}{5}\right)\left(n-\frac{3}{5}\right)$$

Tango \frac{3}{5} mai i n 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.

$$25n^{2}-30n+9=25\times \frac{5n-3}{5}\left(n-\frac{3}{5}\right)$$

Tango \frac{3}{5} mai i n 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.

$$25n^{2}-30n+9=25\times \frac{5n-3}{5}\times \frac{5n-3}{5}$$

Whakareatia \frac{5n-3}{5} ki te \frac{5n-3}{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.

$$25n^{2}-30n+9=25\times \frac{\left(5n-3\right)\left(5n-3\right)}{5\times 5}$$

Whakareatia 5 ki te 5.

$$25n^{2}-30n+9=25\times \frac{\left(5n-3\right)\left(5n-3\right)}{25}$$

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

$$25n^{2}-30n+9=\left(5n-3\right)\left(5n-3\right)$$

Aromātai

$$\left(5n-3\right)^{2}$$