Whakaoti i te 4v^2-12v-16

$$4 v ^ { 2 } - 12 v - 16$$

Tauwehe

$$4\left(v-4\right)\left(v+1\right)$$

Ngā Upane Otinga

Tauwehea te 4.

$$4\left(v^{2}-3v-4\right)$$

Whakaarohia te v^{2}-3v-4. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei v^{2}+av+bv-4. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-3 ab=1\left(-4\right)=-4$$

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 -4.

$$1,-4 2,-2$$

Tātaihia te tapeke mō ia takirua.

$$1-4=-3 2-2=0$$

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

$$a=-4 b=1$$

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

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

Whakatauwehea atu v i te v^{2}-4v.

$$v\left(v-4\right)+v-4$$

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

$$\left(v-4\right)\left(v+1\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$4\left(v-4\right)\left(v+1\right)$$

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.

$$4v^{2}-12v-16=0$$

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.

$$v=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 4\left(-16\right)}}{2\times 4}$$

Pūrua -12.

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

Whakareatia -4 ki te 4.

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

Whakareatia -16 ki te -16.

$$v=\frac{-\left(-12\right)±\sqrt{144+256}}{2\times 4}$$

Tāpiri 144 ki te 256.

$$v=\frac{-\left(-12\right)±\sqrt{400}}{2\times 4}$$

Tuhia te pūtakerua o te 400.

$$v=\frac{-\left(-12\right)±20}{2\times 4}$$

Ko te tauaro o -12 ko 12.

$$v=\frac{12±20}{2\times 4}$$

Whakareatia 2 ki te 4.

$$v=\frac{12±20}{8}$$

Nā, me whakaoti te whārite v=\frac{12±20}{8} ina he tāpiri te ±. Tāpiri 12 ki te 20.

$$v=\frac{32}{8}$$

Whakawehe 32 ki te 8.

$$v=4$$

Nā, me whakaoti te whārite v=\frac{12±20}{8} ina he tango te ±. Tango 20 mai i 12.

$$v=-\frac{8}{8}$$

Whakawehe -8 ki te 8.

$$v=-1$$

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 4 mō te x_{1} me te -1 mō te x_{2}.

$$4v^{2}-12v-16=4\left(v-4\right)\left(v-\left(-1\right)\right)$$

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

$$4v^{2}-12v-16=4\left(v-4\right)\left(v+1\right)$$

Aromātai

$$4\left(v-4\right)\left(v+1\right)$$