Whakaoti i te 4a^2-4a+1

$$4 a ^ { 2 } - 4 a + 1$$

Tauwehe

$$\left(2a-1\right)^{2}$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 4a^{2}+pa+qa+1. Hei kimi p me q, whakaritea tētahi pūnaha kia whakaoti.

$$p+q=-4 pq=4\times 1=4$$

I te mea kua tōrunga te pq, he ōrite te tohu o p me q. I te mea kua tōraro te p+q, he tōraro hoki a p me q. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 4.

$$-1,-4 -2,-2$$

Tātaihia te tapeke mō ia takirua.

$$-1-4=-5 -2-2=-4$$

Ko te otinga te takirua ka hoatu i te tapeke -4.

$$p=-2 q=-2$$

Tuhia anō te 4a^{2}-4a+1 hei \left(4a^{2}-2a\right)+\left(-2a+1\right).

$$\left(4a^{2}-2a\right)+\left(-2a+1\right)$$

Tauwehea te 2a i te tuatahi me te -1 i te rōpū tuarua.

$$2a\left(2a-1\right)-\left(2a-1\right)$$

Whakatauwehea atu te kīanga pātahi 2a-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2a-1\right)\left(2a-1\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(2a-1\right)^{2}$$

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.

$$factor(4a^{2}-4a+1)$$

Kimihia te tauwehe pātahi nui rawa o ngā tau whakarea.

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

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

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

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.

$$\left(2a-1\right)^{2}$$

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.

$$4a^{2}-4a+1=0$$

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.

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

Pūrua -4.

$$a=\frac{-\left(-4\right)±\sqrt{16-4\times 4}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$a=\frac{-\left(-4\right)±\sqrt{16-16}}{2\times 4}$$

Tāpiri 16 ki te -16.

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

Tuhia te pūtakerua o te 0.

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

Ko te tauaro o -4 ko 4.

$$a=\frac{4±0}{2\times 4}$$

Whakareatia 2 ki te 4.

$$a=\frac{4±0}{8}$$

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

$$4a^{2}-4a+1=4\left(a-\frac{1}{2}\right)\left(a-\frac{1}{2}\right)$$

Tango \frac{1}{2} mai i a 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.

$$4a^{2}-4a+1=4\times \frac{2a-1}{2}\left(a-\frac{1}{2}\right)$$

Tango \frac{1}{2} mai i a 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.

$$4a^{2}-4a+1=4\times \frac{2a-1}{2}\times \frac{2a-1}{2}$$

Whakareatia \frac{2a-1}{2} ki te \frac{2a-1}{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.

$$4a^{2}-4a+1=4\times \frac{\left(2a-1\right)\left(2a-1\right)}{2\times 2}$$

Whakareatia 2 ki te 2.

$$4a^{2}-4a+1=4\times \frac{\left(2a-1\right)\left(2a-1\right)}{4}$$

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

$$4a^{2}-4a+1=\left(2a-1\right)\left(2a-1\right)$$

Aromātai

$$\left(2a-1\right)^{2}$$