Whakaoti i te 12x^2+x-6

$$12 x ^ { 2 } + x - 6$$

Tauwehe

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

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 12x^{2}+ax+bx-6. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=1 ab=12\left(-6\right)=-72$$

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 -72.

$$-1,72 -2,36 -3,24 -4,18 -6,12 -8,9$$

Tātaihia te tapeke mō ia takirua.

$$-1+72=71 -2+36=34 -3+24=21 -4+18=14 -6+12=6 -8+9=1$$

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

$$a=-8 b=9$$

Tuhia anō te 12x^{2}+x-6 hei \left(12x^{2}-8x\right)+\left(9x-6\right).

$$\left(12x^{2}-8x\right)+\left(9x-6\right)$$

Tauwehea te 4x i te tuatahi me te 3 i te rōpū tuarua.

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

Whakatauwehea atu te kīanga pātahi 3x-2 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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.

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

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.

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

Pūrua 1.

$$x=\frac{-1±\sqrt{1-4\times 12\left(-6\right)}}{2\times 12}$$

Whakareatia -4 ki te 12.

$$x=\frac{-1±\sqrt{1-48\left(-6\right)}}{2\times 12}$$

Whakareatia -48 ki te -6.

$$x=\frac{-1±\sqrt{1+288}}{2\times 12}$$

Tāpiri 1 ki te 288.

$$x=\frac{-1±\sqrt{289}}{2\times 12}$$

Tuhia te pūtakerua o te 289.

$$x=\frac{-1±17}{2\times 12}$$

Whakareatia 2 ki te 12.

$$x=\frac{-1±17}{24}$$

Nā, me whakaoti te whārite x=\frac{-1±17}{24} ina he tāpiri te ±. Tāpiri -1 ki te 17.

$$x=\frac{16}{24}$$

Whakahekea te hautanga \frac{16}{24} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$x=\frac{2}{3}$$

Nā, me whakaoti te whārite x=\frac{-1±17}{24} ina he tango te ±. Tango 17 mai i -1.

$$x=-\frac{18}{24}$$

Whakahekea te hautanga \frac{-18}{24} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=-\frac{3}{4}$$

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 \frac{2}{3} mō te x_{1} me te -\frac{3}{4} mō te x_{2}.

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

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

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

Tango \frac{2}{3} mai i x mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

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

Tāpiri \frac{3}{4} 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.

$$12x^{2}+x-6=12\times \frac{3x-2}{3}\times \frac{4x+3}{4}$$

Whakareatia \frac{3x-2}{3} ki te \frac{4x+3}{4} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

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

Whakareatia 3 ki te 4.

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

Whakakorea atu te tauwehe pūnoa nui rawa 12 i roto i te 12 me te 12.

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

Aromātai

$$12x^{2}+x-6$$