Whakaoti i te -9x^2-x+10

$$- 9 x ^ { 2 } - x + 10$$

Tauwehe

$$\left(1-x\right)\left(9x+10\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -9x^{2}+ax+bx+10. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=-9\times 10=-90$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -90.

$$1,-90 2,-45 3,-30 5,-18 6,-15 9,-10$$

Tātaihia te tapeke mō ia takirua.

$$1-90=-89 2-45=-43 3-30=-27 5-18=-13 6-15=-9 9-10=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -1.

$$a=9 b=-10$$

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

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

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

$$9x\left(-x+1\right)+10\left(-x+1\right)$$

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

$$\left(-x+1\right)\left(9x+10\right)$$

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.

$$-9x^{2}-x+10=0$$

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.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-9\right)\times 10}}{2\left(-9\right)}$$

Whakareatia -4 ki te -9.

$$x=\frac{-\left(-1\right)±\sqrt{1+36\times 10}}{2\left(-9\right)}$$

Whakareatia 36 ki te 10.

$$x=\frac{-\left(-1\right)±\sqrt{1+360}}{2\left(-9\right)}$$

Tāpiri 1 ki te 360.

$$x=\frac{-\left(-1\right)±\sqrt{361}}{2\left(-9\right)}$$

Tuhia te pūtakerua o te 361.

$$x=\frac{-\left(-1\right)±19}{2\left(-9\right)}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±19}{2\left(-9\right)}$$

Whakareatia 2 ki te -9.

$$x=\frac{1±19}{-18}$$

Nā, me whakaoti te whārite x=\frac{1±19}{-18} ina he tāpiri te ±. Tāpiri 1 ki te 19.

$$x=\frac{20}{-18}$$

Whakahekea te hautanga \frac{20}{-18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{10}{9}$$

Nā, me whakaoti te whārite x=\frac{1±19}{-18} ina he tango te ±. Tango 19 mai i 1.

$$x=-\frac{18}{-18}$$

Whakawehe -18 ki te -18.

$$x=1$$

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 1 mō te x_{2}.

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

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

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

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.

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

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

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

Aromātai

$$\left(1-x\right)\left(9x+10\right)$$