Whakaoti i te 4a^2+28a+48

$$4 a ^ { 2 } + 28 a + 48$$

Tauwehe

$$4\left(a+3\right)\left(a+4\right)$$

Ngā Upane Otinga

Tauwehea te 4.

$$4\left(a^{2}+7a+12\right)$$

Whakaarohia te a^{2}+7a+12. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei a^{2}+pa+qa+12. Hei kimi p me q, whakaritea tētahi pūnaha kia whakaoti.

$$p+q=7 pq=1\times 12=12$$

I te mea kua tōrunga te pq, he ōrite te tohu o p me q. I te mea kua tōrunga te p+q, he tōrunga hoki a p me q. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 12.

$$1,12 2,6 3,4$$

Tātaihia te tapeke mō ia takirua.

$$1+12=13 2+6=8 3+4=7$$

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

$$p=3 q=4$$

Tuhia anō te a^{2}+7a+12 hei \left(a^{2}+3a\right)+\left(4a+12\right).

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

Tauwehea te a i te tuatahi me te 4 i te rōpū tuarua.

$$a\left(a+3\right)+4\left(a+3\right)$$

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

$$\left(a+3\right)\left(a+4\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$4\left(a+3\right)\left(a+4\right)$$

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.

$$4a^{2}+28a+48=0$$

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.

$$a=\frac{-28±\sqrt{28^{2}-4\times 4\times 48}}{2\times 4}$$

Pūrua 28.

$$a=\frac{-28±\sqrt{784-4\times 4\times 48}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$a=\frac{-28±\sqrt{784-16\times 48}}{2\times 4}$$

Whakareatia -16 ki te 48.

$$a=\frac{-28±\sqrt{784-768}}{2\times 4}$$

Tāpiri 784 ki te -768.

$$a=\frac{-28±\sqrt{16}}{2\times 4}$$

Tuhia te pūtakerua o te 16.

$$a=\frac{-28±4}{2\times 4}$$

Whakareatia 2 ki te 4.

$$a=\frac{-28±4}{8}$$

Nā, me whakaoti te whārite a=\frac{-28±4}{8} ina he tāpiri te ±. Tāpiri -28 ki te 4.

$$a=-\frac{24}{8}$$

Whakawehe -24 ki te 8.

$$a=-3$$

Nā, me whakaoti te whārite a=\frac{-28±4}{8} ina he tango te ±. Tango 4 mai i -28.

$$a=-\frac{32}{8}$$

Whakawehe -32 ki te 8.

$$a=-4$$

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 -3 mō te x_{1} me te -4 mō te x_{2}.

$$4a^{2}+28a+48=4\left(a-\left(-3\right)\right)\left(a-\left(-4\right)\right)$$

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

$$4a^{2}+28a+48=4\left(a+3\right)\left(a+4\right)$$

Aromātai

$$4\left(a+3\right)\left(a+4\right)$$