Whakaoti i te -6b^2+b+12?

$$- 6 b ^ { 2 } + b + 12 ?$$

Tauwehe

$$\left(-3b-4\right)\left(2b-3\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -6b^{2}+pb+qb+12. Hei kimi p me q, whakaritea tētahi pūnaha kia whakaoti.

$$p+q=1 pq=-6\times 12=-72$$

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 -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.

$$p=9 q=-8$$

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

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

Tauwehea te -3b i te tuatahi me te -4 i te rōpū tuarua.

$$-3b\left(2b-3\right)-4\left(2b-3\right)$$

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

$$\left(2b-3\right)\left(-3b-4\right)$$

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.

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

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.

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

Pūrua 1.

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

Whakareatia -4 ki te -6.

$$b=\frac{-1±\sqrt{1+24\times 12}}{2\left(-6\right)}$$

Whakareatia 24 ki te 12.

$$b=\frac{-1±\sqrt{1+288}}{2\left(-6\right)}$$

Tāpiri 1 ki te 288.

$$b=\frac{-1±\sqrt{289}}{2\left(-6\right)}$$

Tuhia te pūtakerua o te 289.

$$b=\frac{-1±17}{2\left(-6\right)}$$

Whakareatia 2 ki te -6.

$$b=\frac{-1±17}{-12}$$

Nā, me whakaoti te whārite b=\frac{-1±17}{-12} ina he tāpiri te ±. Tāpiri -1 ki te 17.

$$b=\frac{16}{-12}$$

Whakahekea te hautanga \frac{16}{-12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

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

Nā, me whakaoti te whārite b=\frac{-1±17}{-12} ina he tango te ±. Tango 17 mai i -1.

$$b=-\frac{18}{-12}$$

Whakahekea te hautanga \frac{-18}{-12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

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

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

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

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

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

Tāpiri \frac{4}{3} ki te b 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.

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

Tango \frac{3}{2} mai i b 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.

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

Whakareatia \frac{-3b-4}{-3} ki te \frac{-2b+3}{-2} 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.

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

Whakareatia -3 ki te -2.

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

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

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

Aromātai

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