Whakaoti i te 12z^2-7z-12

$$12 { z }^{ 2 } -7z-12$$

Tauwehe

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

Ngā Upane Otinga

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

$$a+b=-7 ab=12\left(-12\right)=-144$$

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 -144.

$$1,-144 2,-72 3,-48 4,-36 6,-24 8,-18 9,-16 12,-12$$

Tātaihia te tapeke mō ia takirua.

$$1-144=-143 2-72=-70 3-48=-45 4-36=-32 6-24=-18 8-18=-10 9-16=-7 12-12=0$$

Ko te otinga te takirua ka hoatu i te tapeke -7.

$$a=-16 b=9$$

Tuhia anō te 12z^{2}-7z-12 hei \left(12z^{2}-16z\right)+\left(9z-12\right).

$$\left(12z^{2}-16z\right)+\left(9z-12\right)$$

Tauwehea te 4z i te tuatahi me te 3 i te rōpū tuarua.

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

Whakatauwehea atu te kīanga pātahi 3z-4 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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.

$$12z^{2}-7z-12=0$$

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.

$$z=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 12\left(-12\right)}}{2\times 12}$$

Pūrua -7.

$$z=\frac{-\left(-7\right)±\sqrt{49-4\times 12\left(-12\right)}}{2\times 12}$$

Whakareatia -4 ki te 12.

$$z=\frac{-\left(-7\right)±\sqrt{49-48\left(-12\right)}}{2\times 12}$$

Whakareatia -48 ki te -12.

$$z=\frac{-\left(-7\right)±\sqrt{49+576}}{2\times 12}$$

Tāpiri 49 ki te 576.

$$z=\frac{-\left(-7\right)±\sqrt{625}}{2\times 12}$$

Tuhia te pūtakerua o te 625.

$$z=\frac{-\left(-7\right)±25}{2\times 12}$$

Ko te tauaro o -7 ko 7.

$$z=\frac{7±25}{2\times 12}$$

Whakareatia 2 ki te 12.

$$z=\frac{7±25}{24}$$

Nā, me whakaoti te whārite z=\frac{7±25}{24} ina he tāpiri te ±. Tāpiri 7 ki te 25.

$$z=\frac{32}{24}$$

Whakahekea te hautanga \frac{32}{24} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$z=\frac{4}{3}$$

Nā, me whakaoti te whārite z=\frac{7±25}{24} ina he tango te ±. Tango 25 mai i 7.

$$z=-\frac{18}{24}$$

Whakahekea te hautanga \frac{-18}{24} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$z=-\frac{3}{4}$$

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{4}{3} mō te x_{1} me te -\frac{3}{4} mō te x_{2}.

$$12z^{2}-7z-12=12\left(z-\frac{4}{3}\right)\left(z-\left(-\frac{3}{4}\right)\right)$$

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

$$12z^{2}-7z-12=12\left(z-\frac{4}{3}\right)\left(z+\frac{3}{4}\right)$$

Tango \frac{4}{3} mai i z mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$12z^{2}-7z-12=12\times \frac{3z-4}{3}\left(z+\frac{3}{4}\right)$$

Tāpiri \frac{3}{4} 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.

$$12z^{2}-7z-12=12\times \frac{3z-4}{3}\times \frac{4z+3}{4}$$

Whakareatia \frac{3z-4}{3} ki te \frac{4z+3}{4} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$12z^{2}-7z-12=12\times \frac{\left(3z-4\right)\left(4z+3\right)}{3\times 4}$$

Whakareatia 3 ki te 4.

$$12z^{2}-7z-12=12\times \frac{\left(3z-4\right)\left(4z+3\right)}{12}$$

Whakakorea atu te tauwehe pūnoa nui rawa 12 i roto i te 12 me te 12.

$$12z^{2}-7z-12=\left(3z-4\right)\left(4z+3\right)$$

Aromātai

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