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 tōraro te a+b, he tōraro 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.

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

Pūrua -180.

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

Whakareatia -4 ki te 81.

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

Whakareatia -324 ki te 100.

$$x=\frac{-\left(-180\right)±\sqrt{32400-32400}}{2\times 81}$$

Tāpiri 32400 ki te -32400.

$$x=\frac{-\left(-180\right)±\sqrt{0}}{2\times 81}$$

Tuhia te pūtakerua o te 0.

$$x=\frac{-\left(-180\right)±0}{2\times 81}$$

Ko te tauaro o -180 ko 180.

$$x=\frac{180±0}{2\times 81}$$

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}.

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

Tango \frac{10}{9} 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.

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

Tango \frac{10}{9} 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.

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

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

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