Whakaoti i te 4a^2+12a-72

$$4 a ^ { 2 } + 12 a - 72$$

Tauwehe

$$4\left(a-3\right)\left(a+6\right)$$

Ngā Upane Otinga

Tauwehea te 4.

$$4\left(a^{2}+3a-18\right)$$

Whakaarohia te a^{2}+3a-18. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei a^{2}+pa+qa-18. Hei kimi p me q, whakaritea tētahi pūnaha kia whakaoti.

$$p+q=3 pq=1\left(-18\right)=-18$$

I te mea kua tōraro te pq, he tauaro ngā tohu o p me q. I te mea kua tōrunga te p+q, 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 -18.

$$-1,18 -2,9 -3,6$$

Tātaihia te tapeke mō ia takirua.

$$-1+18=17 -2+9=7 -3+6=3$$

Ko te otinga te takirua ka hoatu i te tapeke 3.

$$p=-3 q=6$$

Tuhia anō te a^{2}+3a-18 hei \left(a^{2}-3a\right)+\left(6a-18\right).

$$\left(a^{2}-3a\right)+\left(6a-18\right)$$

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

$$a\left(a-3\right)+6\left(a-3\right)$$

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

$$\left(a-3\right)\left(a+6\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$4\left(a-3\right)\left(a+6\right)$$

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.

$$4a^{2}+12a-72=0$$

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.

$$a=\frac{-12±\sqrt{12^{2}-4\times 4\left(-72\right)}}{2\times 4}$$

Pūrua 12.

$$a=\frac{-12±\sqrt{144-4\times 4\left(-72\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$a=\frac{-12±\sqrt{144-16\left(-72\right)}}{2\times 4}$$

Whakareatia -16 ki te -72.

$$a=\frac{-12±\sqrt{144+1152}}{2\times 4}$$

Tāpiri 144 ki te 1152.

$$a=\frac{-12±\sqrt{1296}}{2\times 4}$$

Tuhia te pūtakerua o te 1296.

$$a=\frac{-12±36}{2\times 4}$$

Whakareatia 2 ki te 4.

$$a=\frac{-12±36}{8}$$

Nā, me whakaoti te whārite a=\frac{-12±36}{8} ina he tāpiri te ±. Tāpiri -12 ki te 36.

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

Whakawehe 24 ki te 8.

$$a=3$$

Nā, me whakaoti te whārite a=\frac{-12±36}{8} ina he tango te ±. Tango 36 mai i -12.

$$a=-\frac{48}{8}$$

Whakawehe -48 ki te 8.

$$a=-6$$

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 3 mō te x_{1} me te -6 mō te x_{2}.

$$4a^{2}+12a-72=4\left(a-3\right)\left(a-\left(-6\right)\right)$$

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

$$4a^{2}+12a-72=4\left(a-3\right)\left(a+6\right)$$

Aromātai

$$4\left(a-3\right)\left(a+6\right)$$