Whakaoti i te 9x^2-9x-108

$$9 x ^ { 2 } - 9 x - 108$$

Tauwehe

$$9\left(x-4\right)\left(x+3\right)$$

Ngā Upane Otinga

Tauwehea te 9.

$$9\left(x^{2}-x-12\right)$$

Whakaarohia te x^{2}-x-12. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei x^{2}+ax+bx-12. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=1\left(-12\right)=-12$$

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 -12.

$$1,-12 2,-6 3,-4$$

Tātaihia te tapeke mō ia takirua.

$$1-12=-11 2-6=-4 3-4=-1$$

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

$$a=-4 b=3$$

Tuhia anō te x^{2}-x-12 hei \left(x^{2}-4x\right)+\left(3x-12\right).

$$\left(x^{2}-4x\right)+\left(3x-12\right)$$

Tauwehea te x i te tuatahi me te 3 i te rōpū tuarua.

$$x\left(x-4\right)+3\left(x-4\right)$$

Whakatauwehea atu te kīanga pātahi x-4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-4\right)\left(x+3\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$9\left(x-4\right)\left(x+3\right)$$

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.

$$9x^{2}-9x-108=0$$

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.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 9\left(-108\right)}}{2\times 9}$$

Pūrua -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 9\left(-108\right)}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$x=\frac{-\left(-9\right)±\sqrt{81-36\left(-108\right)}}{2\times 9}$$

Whakareatia -36 ki te -108.

$$x=\frac{-\left(-9\right)±\sqrt{81+3888}}{2\times 9}$$

Tāpiri 81 ki te 3888.

$$x=\frac{-\left(-9\right)±\sqrt{3969}}{2\times 9}$$

Tuhia te pūtakerua o te 3969.

$$x=\frac{-\left(-9\right)±63}{2\times 9}$$

Ko te tauaro o -9 ko 9.

$$x=\frac{9±63}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=\frac{9±63}{18}$$

Nā, me whakaoti te whārite x=\frac{9±63}{18} ina he tāpiri te ±. Tāpiri 9 ki te 63.

$$x=\frac{72}{18}$$

Whakawehe 72 ki te 18.

$$x=4$$

Nā, me whakaoti te whārite x=\frac{9±63}{18} ina he tango te ±. Tango 63 mai i 9.

$$x=-\frac{54}{18}$$

Whakawehe -54 ki te 18.

$$x=-3$$

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 4 mō te x_{1} me te -3 mō te x_{2}.

$$9x^{2}-9x-108=9\left(x-4\right)\left(x-\left(-3\right)\right)$$

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

$$9x^{2}-9x-108=9\left(x-4\right)\left(x+3\right)$$

Aromātai

$$9\left(x-4\right)\left(x+3\right)$$