Whakaoti i te 4x^2+4x-48

$$4 x ^ { 2 } + 4 x - 48$$

Tauwehe

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

Ngā Upane Otinga

Tauwehea te 4.

$$4\left(x^{2}+x-12\right)$$

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

$$a+b=1 ab=1\left(-12\right)=-12$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, 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 -12.

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

Tātaihia te tapeke mō ia takirua.

$$-1+12=11 -2+6=4 -3+4=1$$

Ko te otinga te takirua ka hoatu i te tapeke 1.

$$a=-3 b=4$$

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

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

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

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

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

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

Me tuhi anō te kīanga whakatauwehe katoa.

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

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.

$$4x^{2}+4x-48=0$$

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.

$$x=\frac{-4±\sqrt{4^{2}-4\times 4\left(-48\right)}}{2\times 4}$$

Pūrua 4.

$$x=\frac{-4±\sqrt{16-4\times 4\left(-48\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-4±\sqrt{16-16\left(-48\right)}}{2\times 4}$$

Whakareatia -16 ki te -48.

$$x=\frac{-4±\sqrt{16+768}}{2\times 4}$$

Tāpiri 16 ki te 768.

$$x=\frac{-4±\sqrt{784}}{2\times 4}$$

Tuhia te pūtakerua o te 784.

$$x=\frac{-4±28}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{-4±28}{8}$$

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

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

Whakawehe 24 ki te 8.

$$x=3$$

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

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

Whakawehe -32 ki te 8.

$$x=-4$$

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}.

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

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

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

Aromātai

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