Whakaoti i te 9c^2-10c+1

$$9 c ^ { 2 } - 10 c + 1$$

Tauwehe

$$\left(c-1\right)\left(9c-1\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 9c^{2}+ac+bc+1. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-10 ab=9\times 1=9$$

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 9.

$$-1,-9 -3,-3$$

Tātaihia te tapeke mō ia takirua.

$$-1-9=-10 -3-3=-6$$

Ko te otinga te takirua ka hoatu i te tapeke -10.

$$a=-9 b=-1$$

Tuhia anō te 9c^{2}-10c+1 hei \left(9c^{2}-9c\right)+\left(-c+1\right).

$$\left(9c^{2}-9c\right)+\left(-c+1\right)$$

Tauwehea te 9c i te tuatahi me te -1 i te rōpū tuarua.

$$9c\left(c-1\right)-\left(c-1\right)$$

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

$$\left(c-1\right)\left(9c-1\right)$$

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.

$$9c^{2}-10c+1=0$$

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.

$$c=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 9}}{2\times 9}$$

Pūrua -10.

$$c=\frac{-\left(-10\right)±\sqrt{100-4\times 9}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$c=\frac{-\left(-10\right)±\sqrt{100-36}}{2\times 9}$$

Tāpiri 100 ki te -36.

$$c=\frac{-\left(-10\right)±\sqrt{64}}{2\times 9}$$

Tuhia te pūtakerua o te 64.

$$c=\frac{-\left(-10\right)±8}{2\times 9}$$

Ko te tauaro o -10 ko 10.

$$c=\frac{10±8}{2\times 9}$$

Whakareatia 2 ki te 9.

$$c=\frac{10±8}{18}$$

Nā, me whakaoti te whārite c=\frac{10±8}{18} ina he tāpiri te ±. Tāpiri 10 ki te 8.

$$c=\frac{18}{18}$$

Whakawehe 18 ki te 18.

$$c=1$$

Nā, me whakaoti te whārite c=\frac{10±8}{18} ina he tango te ±. Tango 8 mai i 10.

$$c=\frac{2}{18}$$

Whakahekea te hautanga \frac{2}{18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$c=\frac{1}{9}$$

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

$$9c^{2}-10c+1=9\left(c-1\right)\left(c-\frac{1}{9}\right)$$

Tango \frac{1}{9} mai i c 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.

$$9c^{2}-10c+1=9\left(c-1\right)\times \frac{9c-1}{9}$$

Whakakorea atu te tauwehe pūnoa nui rawa 9 i roto i te 9 me te 9.

$$9c^{2}-10c+1=\left(c-1\right)\left(9c-1\right)$$

Aromātai

$$\left(c-1\right)\left(9c-1\right)$$