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

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

Tauwehe

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

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 81x^{2}+ax+bx+25. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=90 ab=81\times 25=2025$$

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 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.

$$a=45 b=45$$

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

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

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

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

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

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

Tuhia anōtia hei pūrua huarua.

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

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.

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

Kimihia te tauwehe pātahi nui rawa o ngā tau whakarea.

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

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

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

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.

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

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.

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

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.

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

Pūrua 90.

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

Whakareatia -4 ki te 81.

$$x=\frac{-90±\sqrt{8100-324\times 25}}{2\times 81}$$

Whakareatia -324 ki te 25.

$$x=\frac{-90±\sqrt{8100-8100}}{2\times 81}$$

Tāpiri 8100 ki te -8100.

$$x=\frac{-90±\sqrt{0}}{2\times 81}$$

Tuhia te pūtakerua o te 0.

$$x=\frac{-90±0}{2\times 81}$$

Whakareatia 2 ki te 81.

$$x=\frac{-90±0}{162}$$

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

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

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

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

Tāpiri \frac{5}{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.

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

Tāpiri \frac{5}{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.

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

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

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

Whakareatia 9 ki te 9.

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

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

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

Aromātai

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