Whakaoti i te 2z^2+19z-21

$$2 z ^ { 2 } + 19 z - 21$$

Tauwehe

$$\left(z-1\right)\left(2z+21\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 2z^{2}+az+bz-21. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=19 ab=2\left(-21\right)=-42$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -42.

$$-1,42 -2,21 -3,14 -6,7$$

Tātaihia te tapeke mō ia takirua.

$$-1+42=41 -2+21=19 -3+14=11 -6+7=1$$

Ko te otinga te takirua ka hoatu i te tapeke 19.

$$a=-2 b=21$$

Tuhia anō te 2z^{2}+19z-21 hei \left(2z^{2}-2z\right)+\left(21z-21\right).

$$\left(2z^{2}-2z\right)+\left(21z-21\right)$$

Tauwehea te 2z i te tuatahi me te 21 i te rōpū tuarua.

$$2z\left(z-1\right)+21\left(z-1\right)$$

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

$$\left(z-1\right)\left(2z+21\right)$$

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.

$$2z^{2}+19z-21=0$$

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.

$$z=\frac{-19±\sqrt{19^{2}-4\times 2\left(-21\right)}}{2\times 2}$$

Pūrua 19.

$$z=\frac{-19±\sqrt{361-4\times 2\left(-21\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$z=\frac{-19±\sqrt{361-8\left(-21\right)}}{2\times 2}$$

Whakareatia -8 ki te -21.

$$z=\frac{-19±\sqrt{361+168}}{2\times 2}$$

Tāpiri 361 ki te 168.

$$z=\frac{-19±\sqrt{529}}{2\times 2}$$

Tuhia te pūtakerua o te 529.

$$z=\frac{-19±23}{2\times 2}$$

Whakareatia 2 ki te 2.

$$z=\frac{-19±23}{4}$$

Nā, me whakaoti te whārite z=\frac{-19±23}{4} ina he tāpiri te ±. Tāpiri -19 ki te 23.

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

Whakawehe 4 ki te 4.

$$z=1$$

Nā, me whakaoti te whārite z=\frac{-19±23}{4} ina he tango te ±. Tango 23 mai i -19.

$$z=-\frac{42}{4}$$

Whakahekea te hautanga \frac{-42}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$z=-\frac{21}{2}$$

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 1 mō te x_{1} me te -\frac{21}{2} mō te x_{2}.

$$2z^{2}+19z-21=2\left(z-1\right)\left(z-\left(-\frac{21}{2}\right)\right)$$

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

$$2z^{2}+19z-21=2\left(z-1\right)\left(z+\frac{21}{2}\right)$$

Tāpiri \frac{21}{2} ki te z 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.

$$2z^{2}+19z-21=2\left(z-1\right)\times \frac{2z+21}{2}$$

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

$$2z^{2}+19z-21=\left(z-1\right)\left(2z+21\right)$$

Aromātai

$$\left(z-1\right)\left(2z+21\right)$$