Whakaoti i te 10z^2+21z+2

$$10 z ^ { 2 } + 21 z + 2$$

Tauwehe

$$\left(z+2\right)\left(10z+1\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 10z^{2}+az+bz+2. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=21 ab=10\times 2=20$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 20.

$$1,20 2,10 4,5$$

Tātaihia te tapeke mō ia takirua.

$$1+20=21 2+10=12 4+5=9$$

Ko te otinga te takirua ka hoatu i te tapeke 21.

$$a=1 b=20$$

Tuhia anō te 10z^{2}+21z+2 hei \left(10z^{2}+z\right)+\left(20z+2\right).

$$\left(10z^{2}+z\right)+\left(20z+2\right)$$

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

$$z\left(10z+1\right)+2\left(10z+1\right)$$

Whakatauwehea atu te kīanga pātahi 10z+1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(10z+1\right)\left(z+2\right)$$

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.

$$10z^{2}+21z+2=0$$

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.

$$z=\frac{-21±\sqrt{21^{2}-4\times 10\times 2}}{2\times 10}$$

Pūrua 21.

$$z=\frac{-21±\sqrt{441-4\times 10\times 2}}{2\times 10}$$

Whakareatia -4 ki te 10.

$$z=\frac{-21±\sqrt{441-40\times 2}}{2\times 10}$$

Whakareatia -40 ki te 2.

$$z=\frac{-21±\sqrt{441-80}}{2\times 10}$$

Tāpiri 441 ki te -80.

$$z=\frac{-21±\sqrt{361}}{2\times 10}$$

Tuhia te pūtakerua o te 361.

$$z=\frac{-21±19}{2\times 10}$$

Whakareatia 2 ki te 10.

$$z=\frac{-21±19}{20}$$

Nā, me whakaoti te whārite z=\frac{-21±19}{20} ina he tāpiri te ±. Tāpiri -21 ki te 19.

$$z=-\frac{2}{20}$$

Whakahekea te hautanga \frac{-2}{20} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$z=-\frac{1}{10}$$

Nā, me whakaoti te whārite z=\frac{-21±19}{20} ina he tango te ±. Tango 19 mai i -21.

$$z=-\frac{40}{20}$$

Whakawehe -40 ki te 20.

$$z=-2$$

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 -\frac{1}{10} mō te x_{1} me te -2 mō te x_{2}.

$$10z^{2}+21z+2=10\left(z-\left(-\frac{1}{10}\right)\right)\left(z-\left(-2\right)\right)$$

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

$$10z^{2}+21z+2=10\left(z+\frac{1}{10}\right)\left(z+2\right)$$

Tāpiri \frac{1}{10} 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.

$$10z^{2}+21z+2=10\times \frac{10z+1}{10}\left(z+2\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 10 i roto i te 10 me te 10.

$$10z^{2}+21z+2=\left(10z+1\right)\left(z+2\right)$$

Aromātai

$$\left(z+2\right)\left(10z+1\right)$$