Whakaoti i te 25b^2+90b+81

$$25 b ^ { 2 } + 90 b + 81$$

Tauwehe

$$\left(5b+9\right)^{2}$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 25b^{2}+pb+qb+81. Hei kimi p me q, whakaritea tētahi pūnaha kia whakaoti.

$$p+q=90 pq=25\times 81=2025$$

I te mea kua tōrunga te pq, he ōrite te tohu o p me q. I te mea kua tōrunga te p+q, he tōrunga hoki a p me q. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 2025.

$$1,2025 3,675 5,405 9,225 15,135 25,81 27,75 45,45$$

Tātaihia te tapeke mō ia takirua.

$$1+2025=2026 3+675=678 5+405=410 9+225=234 15+135=150 25+81=106 27+75=102 45+45=90$$

Ko te otinga te takirua ka hoatu i te tapeke 90.

$$p=45 q=45$$

Tuhia anō te 25b^{2}+90b+81 hei \left(25b^{2}+45b\right)+\left(45b+81\right).

$$\left(25b^{2}+45b\right)+\left(45b+81\right)$$

Tauwehea te 5b i te tuatahi me te 9 i te rōpū tuarua.

$$5b\left(5b+9\right)+9\left(5b+9\right)$$

Whakatauwehea atu te kīanga pātahi 5b+9 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(5b+9\right)\left(5b+9\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(5b+9\right)^{2}$$

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.

$$factor(25b^{2}+90b+81)$$

Kimihia te tauwehe pātahi nui rawa o ngā tau whakarea.

$$gcf(25,90,81)=1$$

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

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

Kimihia te pūtakerua o te kīanga tau autō, 81.

$$\sqrt{81}=9$$

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.

$$\left(5b+9\right)^{2}$$

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.

$$25b^{2}+90b+81=0$$

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.

$$b=\frac{-90±\sqrt{90^{2}-4\times 25\times 81}}{2\times 25}$$

Pūrua 90.

$$b=\frac{-90±\sqrt{8100-4\times 25\times 81}}{2\times 25}$$

Whakareatia -4 ki te 25.

$$b=\frac{-90±\sqrt{8100-100\times 81}}{2\times 25}$$

Whakareatia -100 ki te 81.

$$b=\frac{-90±\sqrt{8100-8100}}{2\times 25}$$

Tāpiri 8100 ki te -8100.

$$b=\frac{-90±\sqrt{0}}{2\times 25}$$

Tuhia te pūtakerua o te 0.

$$b=\frac{-90±0}{2\times 25}$$

Whakareatia 2 ki te 25.

$$b=\frac{-90±0}{50}$$

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{9}{5} mō te x_{1} me te -\frac{9}{5} mō te x_{2}.

$$25b^{2}+90b+81=25\left(b-\left(-\frac{9}{5}\right)\right)\left(b-\left(-\frac{9}{5}\right)\right)$$

Whakamāmātia ngā kīanga katoa o te āhua p-\left(-q\right) ki te p+q.

$$25b^{2}+90b+81=25\left(b+\frac{9}{5}\right)\left(b+\frac{9}{5}\right)$$

Tāpiri \frac{9}{5} ki te b mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$25b^{2}+90b+81=25\times \frac{5b+9}{5}\left(b+\frac{9}{5}\right)$$

Tāpiri \frac{9}{5} ki te b mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$25b^{2}+90b+81=25\times \frac{5b+9}{5}\times \frac{5b+9}{5}$$

Whakareatia \frac{5b+9}{5} ki te \frac{5b+9}{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.

$$25b^{2}+90b+81=25\times \frac{\left(5b+9\right)\left(5b+9\right)}{5\times 5}$$

Whakareatia 5 ki te 5.

$$25b^{2}+90b+81=25\times \frac{\left(5b+9\right)\left(5b+9\right)}{25}$$

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

$$25b^{2}+90b+81=\left(5b+9\right)\left(5b+9\right)$$

Aromātai

$$\left(5b+9\right)^{2}$$