Whakaoti i te 12g^2+80g+48

$$12 g ^ { 2 } + 80 g + 48$$

Tauwehe

$$4\left(g+6\right)\left(3g+2\right)$$

Ngā Upane Otinga

Tauwehea te 4.

$$4\left(3g^{2}+20g+12\right)$$

Whakaarohia te 3g^{2}+20g+12. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 3g^{2}+ag+bg+12. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=20 ab=3\times 12=36$$

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 36.

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

Tātaihia te tapeke mō ia takirua.

$$1+36=37 2+18=20 3+12=15 4+9=13 6+6=12$$

Ko te otinga te takirua ka hoatu i te tapeke 20.

$$a=2 b=18$$

Tuhia anō te 3g^{2}+20g+12 hei \left(3g^{2}+2g\right)+\left(18g+12\right).

$$\left(3g^{2}+2g\right)+\left(18g+12\right)$$

Tauwehea te g i te tuatahi me te 6 i te rōpū tuarua.

$$g\left(3g+2\right)+6\left(3g+2\right)$$

Whakatauwehea atu te kīanga pātahi 3g+2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3g+2\right)\left(g+6\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$4\left(3g+2\right)\left(g+6\right)$$

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.

$$12g^{2}+80g+48=0$$

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.

$$g=\frac{-80±\sqrt{80^{2}-4\times 12\times 48}}{2\times 12}$$

Pūrua 80.

$$g=\frac{-80±\sqrt{6400-4\times 12\times 48}}{2\times 12}$$

Whakareatia -4 ki te 12.

$$g=\frac{-80±\sqrt{6400-48\times 48}}{2\times 12}$$

Whakareatia -48 ki te 48.

$$g=\frac{-80±\sqrt{6400-2304}}{2\times 12}$$

Tāpiri 6400 ki te -2304.

$$g=\frac{-80±\sqrt{4096}}{2\times 12}$$

Tuhia te pūtakerua o te 4096.

$$g=\frac{-80±64}{2\times 12}$$

Whakareatia 2 ki te 12.

$$g=\frac{-80±64}{24}$$

Nā, me whakaoti te whārite g=\frac{-80±64}{24} ina he tāpiri te ±. Tāpiri -80 ki te 64.

$$g=-\frac{16}{24}$$

Whakahekea te hautanga \frac{-16}{24} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$g=-\frac{2}{3}$$

Nā, me whakaoti te whārite g=\frac{-80±64}{24} ina he tango te ±. Tango 64 mai i -80.

$$g=-\frac{144}{24}$$

Whakawehe -144 ki te 24.

$$g=-6$$

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{2}{3} mō te x_{1} me te -6 mō te x_{2}.

$$12g^{2}+80g+48=12\left(g-\left(-\frac{2}{3}\right)\right)\left(g-\left(-6\right)\right)$$

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

$$12g^{2}+80g+48=12\left(g+\frac{2}{3}\right)\left(g+6\right)$$

Tāpiri \frac{2}{3} ki te g 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.

$$12g^{2}+80g+48=12\times \frac{3g+2}{3}\left(g+6\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 3 i roto i te 12 me te 3.

$$12g^{2}+80g+48=4\left(3g+2\right)\left(g+6\right)$$

Aromātai

$$4\left(g+6\right)\left(3g+2\right)$$