Whakaoti i te 81x^2+180x+100

$$81 x ^ { 2 } + 180 x + 100$$

Tauwehe

$$\left(9x+10\right)^{2}$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 81x^{2}+ax+bx+100. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=180 ab=81\times 100=8100$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 8100.

$$1,8100 2,4050 3,2700 4,2025 5,1620 6,1350 9,900 10,810 12,675 15,540 18,450 20,405 25,324 27,300 30,270 36,225 45,180 50,162 54,150 60,135 75,108 81,100 90,90$$

Tātaihia te tapeke mō ia takirua.

$$1+8100=8101 2+4050=4052 3+2700=2703 4+2025=2029 5+1620=1625 6+1350=1356 9+900=909 10+810=820 12+675=687 15+540=555 18+450=468 20+405=425 25+324=349 27+300=327 30+270=300 36+225=261 45+180=225 50+162=212 54+150=204 60+135=195 75+108=183 81+100=181 90+90=180$$

Ko te otinga te takirua ka hoatu i te tapeke 180.

$$a=90 b=90$$

Tuhia anō te 81x^{2}+180x+100 hei \left(81x^{2}+90x\right)+\left(90x+100\right).

$$\left(81x^{2}+90x\right)+\left(90x+100\right)$$

Tauwehea te 9x i te tuatahi me te 10 i te rōpū tuarua.

$$9x\left(9x+10\right)+10\left(9x+10\right)$$

Whakatauwehea atu te kīanga pātahi 9x+10 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(9x+10\right)\left(9x+10\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(9x+10\right)^{2}$$

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.

$$factor(81x^{2}+180x+100)$$

Kimihia te tauwehe pātahi nui rawa o ngā tau whakarea.

$$gcf(81,180,100)=1$$

Kimihia te pūtakerua o te kīanga tau ārahi, 81x^{2}.

$$\sqrt{81x^{2}}=9x$$

Kimihia te pūtakerua o te kīanga tau autō, 100.

$$\sqrt{100}=10$$

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.

$$\left(9x+10\right)^{2}$$

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.

$$81x^{2}+180x+100=0$$

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.

$$x=\frac{-180±\sqrt{180^{2}-4\times 81\times 100}}{2\times 81}$$

Pūrua 180.

$$x=\frac{-180±\sqrt{32400-4\times 81\times 100}}{2\times 81}$$

Whakareatia -4 ki te 81.

$$x=\frac{-180±\sqrt{32400-324\times 100}}{2\times 81}$$

Whakareatia -324 ki te 100.

$$x=\frac{-180±\sqrt{32400-32400}}{2\times 81}$$

Tāpiri 32400 ki te -32400.

$$x=\frac{-180±\sqrt{0}}{2\times 81}$$

Tuhia te pūtakerua o te 0.

$$x=\frac{-180±0}{2\times 81}$$

Whakareatia 2 ki te 81.

$$x=\frac{-180±0}{162}$$

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{10}{9} mō te x_{1} me te -\frac{10}{9} mō te x_{2}.

$$81x^{2}+180x+100=81\left(x-\left(-\frac{10}{9}\right)\right)\left(x-\left(-\frac{10}{9}\right)\right)$$

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

$$81x^{2}+180x+100=81\left(x+\frac{10}{9}\right)\left(x+\frac{10}{9}\right)$$

Tāpiri \frac{10}{9} ki te x 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.

$$81x^{2}+180x+100=81\times \frac{9x+10}{9}\left(x+\frac{10}{9}\right)$$

Tāpiri \frac{10}{9} ki te x 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.

$$81x^{2}+180x+100=81\times \frac{9x+10}{9}\times \frac{9x+10}{9}$$

Whakareatia \frac{9x+10}{9} ki te \frac{9x+10}{9} 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.

$$81x^{2}+180x+100=81\times \frac{\left(9x+10\right)\left(9x+10\right)}{9\times 9}$$

Whakareatia 9 ki te 9.

$$81x^{2}+180x+100=81\times \frac{\left(9x+10\right)\left(9x+10\right)}{81}$$

Whakakorea atu te tauwehe pūnoa nui rawa 81 i roto i te 81 me te 81.

$$81x^{2}+180x+100=\left(9x+10\right)\left(9x+10\right)$$

Aromātai

$$\left(9x+10\right)^{2}$$