Whakaoti i te (6x^2-21x-12)

$$( 6 x ^ { 2 } - 21 x - 12 )$$

Tauwehe

$$3\left(x-4\right)\left(2x+1\right)$$

Ngā Upane Otinga

Tauwehea te 3.

$$3\left(2x^{2}-7x-4\right)$$

Whakaarohia te 2x^{2}-7x-4. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 2x^{2}+ax+bx-4. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-7 ab=2\left(-4\right)=-8$$

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 -8.

$$1,-8 2,-4$$

Tātaihia te tapeke mō ia takirua.

$$1-8=-7 2-4=-2$$

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

$$a=-8 b=1$$

Tuhia anō te 2x^{2}-7x-4 hei \left(2x^{2}-8x\right)+\left(x-4\right).

$$\left(2x^{2}-8x\right)+\left(x-4\right)$$

Whakatauwehea atu 2x i te 2x^{2}-8x.

$$2x\left(x-4\right)+x-4$$

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

$$\left(x-4\right)\left(2x+1\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$3\left(x-4\right)\left(2x+1\right)$$

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.

$$6x^{2}-21x-12=0$$

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.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 6\left(-12\right)}}{2\times 6}$$

Pūrua -21.

$$x=\frac{-\left(-21\right)±\sqrt{441-4\times 6\left(-12\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-\left(-21\right)±\sqrt{441-24\left(-12\right)}}{2\times 6}$$

Whakareatia -24 ki te -12.

$$x=\frac{-\left(-21\right)±\sqrt{441+288}}{2\times 6}$$

Tāpiri 441 ki te 288.

$$x=\frac{-\left(-21\right)±\sqrt{729}}{2\times 6}$$

Tuhia te pūtakerua o te 729.

$$x=\frac{-\left(-21\right)±27}{2\times 6}$$

Ko te tauaro o -21 ko 21.

$$x=\frac{21±27}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{21±27}{12}$$

Nā, me whakaoti te whārite x=\frac{21±27}{12} ina he tāpiri te ±. Tāpiri 21 ki te 27.

$$x=\frac{48}{12}$$

Whakawehe 48 ki te 12.

$$x=4$$

Nā, me whakaoti te whārite x=\frac{21±27}{12} ina he tango te ±. Tango 27 mai i 21.

$$x=-\frac{6}{12}$$

Whakahekea te hautanga \frac{-6}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=-\frac{1}{2}$$

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 -\frac{1}{2} mō te x_{2}.

$$6x^{2}-21x-12=6\left(x-4\right)\left(x-\left(-\frac{1}{2}\right)\right)$$

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

$$6x^{2}-21x-12=6\left(x-4\right)\left(x+\frac{1}{2}\right)$$

Tāpiri \frac{1}{2} ki te x 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.

$$6x^{2}-21x-12=6\left(x-4\right)\times \frac{2x+1}{2}$$

Whakakorea atu te tauwehe pūnoa nui rawa 2 i roto i te 6 me te 2.

$$6x^{2}-21x-12=3\left(x-4\right)\left(2x+1\right)$$

Aromātai

$$3\left(x-4\right)\left(2x+1\right)$$