Whakaoti i te x^2-9x-36

$$x ^ { 2 } - 9 x - 36$$

Tauwehe

$$\left(x-12\right)\left(x+3\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei x^{2}+ax+bx-36. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-9 ab=1\left(-36\right)=-36$$

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 -36.

$$1,-36 2,-18 3,-12 4,-9 6,-6$$

Tātaihia te tapeke mō ia takirua.

$$1-36=-35 2-18=-16 3-12=-9 4-9=-5 6-6=0$$

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

$$a=-12 b=3$$

Tuhia anō te x^{2}-9x-36 hei \left(x^{2}-12x\right)+\left(3x-36\right).

$$\left(x^{2}-12x\right)+\left(3x-36\right)$$

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

$$x\left(x-12\right)+3\left(x-12\right)$$

Whakatauwehea atu te kīanga pātahi x-12 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-12\right)\left(x+3\right)$$

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.

$$x^{2}-9x-36=0$$

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.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\left(-36\right)}}{2}$$

Pūrua -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\left(-36\right)}}{2}$$

Whakareatia -4 ki te -36.

$$x=\frac{-\left(-9\right)±\sqrt{81+144}}{2}$$

Tāpiri 81 ki te 144.

$$x=\frac{-\left(-9\right)±\sqrt{225}}{2}$$

Tuhia te pūtakerua o te 225.

$$x=\frac{-\left(-9\right)±15}{2}$$

Ko te tauaro o -9 ko 9.

$$x=\frac{9±15}{2}$$

Nā, me whakaoti te whārite x=\frac{9±15}{2} ina he tāpiri te ±. Tāpiri 9 ki te 15.

$$x=\frac{24}{2}$$

Whakawehe 24 ki te 2.

$$x=12$$

Nā, me whakaoti te whārite x=\frac{9±15}{2} ina he tango te ±. Tango 15 mai i 9.

$$x=-\frac{6}{2}$$

Whakawehe -6 ki te 2.

$$x=-3$$

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 12 mō te x_{1} me te -3 mō te x_{2}.

$$x^{2}-9x-36=\left(x-12\right)\left(x-\left(-3\right)\right)$$

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

$$x^{2}-9x-36=\left(x-12\right)\left(x+3\right)$$

Aromātai

$$\left(x-12\right)\left(x+3\right)$$