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

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

Tauwehe

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

Ngā Upane Otinga

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

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

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 9x^{2}-30x+25 hei \left(9x^{2}-15x\right)+\left(-15x+25\right).

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

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

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

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

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

Tuhia anōtia hei pūrua huarua.

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

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.

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

Kimihia te tauwehe pātahi nui rawa o ngā tau whakarea.

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

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

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

Kimihia te pūtakerua o te kīanga tau autō, 25.

$$\sqrt{25}=5$$

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.

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

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.

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

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.

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

Pūrua -30.

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

Whakareatia -4 ki te 9.

$$x=\frac{-\left(-30\right)±\sqrt{900-36\times 25}}{2\times 9}$$

Whakareatia -36 ki te 25.

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

Tāpiri 900 ki te -900.

$$x=\frac{-\left(-30\right)±\sqrt{0}}{2\times 9}$$

Tuhia te pūtakerua o te 0.

$$x=\frac{-\left(-30\right)±0}{2\times 9}$$

Ko te tauaro o -30 ko 30.

$$x=\frac{30±0}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=\frac{30±0}{18}$$

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

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

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

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

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

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

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

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

Whakareatia 3 ki te 3.

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

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

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

Aromātai

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