Whakaoti i te z^2-6z-16

$$z ^ { 2 } - 6 z - 16$$

Tauwehe

$$\left(z-8\right)\left(z+2\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei z^{2}+az+bz-16. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-6 ab=1\left(-16\right)=-16$$

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 -16.

$$1,-16 2,-8 4,-4$$

Tātaihia te tapeke mō ia takirua.

$$1-16=-15 2-8=-6 4-4=0$$

Ko te otinga te takirua ka hoatu i te tapeke -6.

$$a=-8 b=2$$

Tuhia anō te z^{2}-6z-16 hei \left(z^{2}-8z\right)+\left(2z-16\right).

$$\left(z^{2}-8z\right)+\left(2z-16\right)$$

Tauwehea te z i te tuatahi me te 2 i te rōpū tuarua.

$$z\left(z-8\right)+2\left(z-8\right)$$

Whakatauwehea atu te kīanga pātahi z-8 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(z-8\right)\left(z+2\right)$$

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.

$$z^{2}-6z-16=0$$

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.

$$z=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-16\right)}}{2}$$

Pūrua -6.

$$z=\frac{-\left(-6\right)±\sqrt{36-4\left(-16\right)}}{2}$$

Whakareatia -4 ki te -16.

$$z=\frac{-\left(-6\right)±\sqrt{36+64}}{2}$$

Tāpiri 36 ki te 64.

$$z=\frac{-\left(-6\right)±\sqrt{100}}{2}$$

Tuhia te pūtakerua o te 100.

$$z=\frac{-\left(-6\right)±10}{2}$$

Ko te tauaro o -6 ko 6.

$$z=\frac{6±10}{2}$$

Nā, me whakaoti te whārite z=\frac{6±10}{2} ina he tāpiri te ±. Tāpiri 6 ki te 10.

$$z=\frac{16}{2}$$

Whakawehe 16 ki te 2.

$$z=8$$

Nā, me whakaoti te whārite z=\frac{6±10}{2} ina he tango te ±. Tango 10 mai i 6.

$$z=-\frac{4}{2}$$

Whakawehe -4 ki te 2.

$$z=-2$$

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 8 mō te x_{1} me te -2 mō te x_{2}.

$$z^{2}-6z-16=\left(z-8\right)\left(z-\left(-2\right)\right)$$

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

$$z^{2}-6z-16=\left(z-8\right)\left(z+2\right)$$

Aromātai

$$\left(z-8\right)\left(z+2\right)$$