Whakaoti i te 6b^2-27b-15

$$6 b ^ { 2 } - 27 b - 15$$

Tauwehe

$$3\left(b-5\right)\left(2b+1\right)$$

Ngā Upane Otinga

Tauwehea te 3.

$$3\left(2b^{2}-9b-5\right)$$

Whakaarohia te 2b^{2}-9b-5. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 2b^{2}+pb+qb-5. Hei kimi p me q, whakaritea tētahi pūnaha kia whakaoti.

$$p+q=-9 pq=2\left(-5\right)=-10$$

I te mea kua tōraro te pq, he tauaro ngā tohu o p me q. I te mea kua tōraro te p+q, 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 -10.

$$1,-10 2,-5$$

Tātaihia te tapeke mō ia takirua.

$$1-10=-9 2-5=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -9.

$$p=-10 q=1$$

Tuhia anō te 2b^{2}-9b-5 hei \left(2b^{2}-10b\right)+\left(b-5\right).

$$\left(2b^{2}-10b\right)+\left(b-5\right)$$

Whakatauwehea atu 2b i te 2b^{2}-10b.

$$2b\left(b-5\right)+b-5$$

Whakatauwehea atu te kīanga pātahi b-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(b-5\right)\left(2b+1\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$3\left(b-5\right)\left(2b+1\right)$$

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.

$$6b^{2}-27b-15=0$$

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.

$$b=\frac{-\left(-27\right)±\sqrt{\left(-27\right)^{2}-4\times 6\left(-15\right)}}{2\times 6}$$

Pūrua -27.

$$b=\frac{-\left(-27\right)±\sqrt{729-4\times 6\left(-15\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$b=\frac{-\left(-27\right)±\sqrt{729-24\left(-15\right)}}{2\times 6}$$

Whakareatia -24 ki te -15.

$$b=\frac{-\left(-27\right)±\sqrt{729+360}}{2\times 6}$$

Tāpiri 729 ki te 360.

$$b=\frac{-\left(-27\right)±\sqrt{1089}}{2\times 6}$$

Tuhia te pūtakerua o te 1089.

$$b=\frac{-\left(-27\right)±33}{2\times 6}$$

Ko te tauaro o -27 ko 27.

$$b=\frac{27±33}{2\times 6}$$

Whakareatia 2 ki te 6.

$$b=\frac{27±33}{12}$$

Nā, me whakaoti te whārite b=\frac{27±33}{12} ina he tāpiri te ±. Tāpiri 27 ki te 33.

$$b=\frac{60}{12}$$

Whakawehe 60 ki te 12.

$$b=5$$

Nā, me whakaoti te whārite b=\frac{27±33}{12} ina he tango te ±. Tango 33 mai i 27.

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

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

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

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

$$6b^{2}-27b-15=6\left(b-5\right)\left(b-\left(-\frac{1}{2}\right)\right)$$

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

$$6b^{2}-27b-15=6\left(b-5\right)\left(b+\frac{1}{2}\right)$$

Tāpiri \frac{1}{2} 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.

$$6b^{2}-27b-15=6\left(b-5\right)\times \frac{2b+1}{2}$$

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

$$6b^{2}-27b-15=3\left(b-5\right)\left(2b+1\right)$$

Aromātai

$$3\left(b-5\right)\left(2b+1\right)$$