Whakaoti i te 2x^2-9x-81

$$2 x ^ { 2 } - 9 x - 81$$

Tauwehe

$$\left(x-9\right)\left(2x+9\right)$$

Ngā Upane Otinga

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

$$a+b=-9 ab=2\left(-81\right)=-162$$

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 -162.

$$1,-162 2,-81 3,-54 6,-27 9,-18$$

Tātaihia te tapeke mō ia takirua.

$$1-162=-161 2-81=-79 3-54=-51 6-27=-21 9-18=-9$$

Ko te otinga te takirua ka hoatu i te tapeke -9.

$$a=-18 b=9$$

Tuhia anō te 2x^{2}-9x-81 hei \left(2x^{2}-18x\right)+\left(9x-81\right).

$$\left(2x^{2}-18x\right)+\left(9x-81\right)$$

Tauwehea te 2x i te tuatahi me te 9 i te rōpū tuarua.

$$2x\left(x-9\right)+9\left(x-9\right)$$

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

$$\left(x-9\right)\left(2x+9\right)$$

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.

$$2x^{2}-9x-81=0$$

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.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 2\left(-81\right)}}{2\times 2}$$

Pūrua -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 2\left(-81\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-9\right)±\sqrt{81-8\left(-81\right)}}{2\times 2}$$

Whakareatia -8 ki te -81.

$$x=\frac{-\left(-9\right)±\sqrt{81+648}}{2\times 2}$$

Tāpiri 81 ki te 648.

$$x=\frac{-\left(-9\right)±\sqrt{729}}{2\times 2}$$

Tuhia te pūtakerua o te 729.

$$x=\frac{-\left(-9\right)±27}{2\times 2}$$

Ko te tauaro o -9 ko 9.

$$x=\frac{9±27}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{9±27}{4}$$

Nā, me whakaoti te whārite x=\frac{9±27}{4} ina he tāpiri te ±. Tāpiri 9 ki te 27.

$$x=\frac{36}{4}$$

Whakawehe 36 ki te 4.

$$x=9$$

Nā, me whakaoti te whārite x=\frac{9±27}{4} ina he tango te ±. Tango 27 mai i 9.

$$x=-\frac{18}{4}$$

Whakahekea te hautanga \frac{-18}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{9}{2}$$

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

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

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

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

Tāpiri \frac{9}{2} 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.

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

Whakakorea atu te tauwehe pūnoa nui rawa 2 i roto i te 2 me te 2.

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

Aromātai

$$\left(x-9\right)\left(2x+9\right)$$