Whakaoti i te 9b^2-3b-56

$$9 b ^ { 2 } - 3 b - 56$$

Tauwehe

$$\left(3b-8\right)\left(3b+7\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 9b^{2}+pb+qb-56. Hei kimi p me q, whakaritea tētahi pūnaha kia whakaoti.

$$p+q=-3 pq=9\left(-56\right)=-504$$

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 -504.

$$1,-504 2,-252 3,-168 4,-126 6,-84 7,-72 8,-63 9,-56 12,-42 14,-36 18,-28 21,-24$$

Tātaihia te tapeke mō ia takirua.

$$1-504=-503 2-252=-250 3-168=-165 4-126=-122 6-84=-78 7-72=-65 8-63=-55 9-56=-47 12-42=-30 14-36=-22 18-28=-10 21-24=-3$$

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

$$p=-24 q=21$$

Tuhia anō te 9b^{2}-3b-56 hei \left(9b^{2}-24b\right)+\left(21b-56\right).

$$\left(9b^{2}-24b\right)+\left(21b-56\right)$$

Tauwehea te 3b i te tuatahi me te 7 i te rōpū tuarua.

$$3b\left(3b-8\right)+7\left(3b-8\right)$$

Whakatauwehea atu te kīanga pātahi 3b-8 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3b-8\right)\left(3b+7\right)$$

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.

$$9b^{2}-3b-56=0$$

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.

$$b=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 9\left(-56\right)}}{2\times 9}$$

Pūrua -3.

$$b=\frac{-\left(-3\right)±\sqrt{9-4\times 9\left(-56\right)}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$b=\frac{-\left(-3\right)±\sqrt{9-36\left(-56\right)}}{2\times 9}$$

Whakareatia -36 ki te -56.

$$b=\frac{-\left(-3\right)±\sqrt{9+2016}}{2\times 9}$$

Tāpiri 9 ki te 2016.

$$b=\frac{-\left(-3\right)±\sqrt{2025}}{2\times 9}$$

Tuhia te pūtakerua o te 2025.

$$b=\frac{-\left(-3\right)±45}{2\times 9}$$

Ko te tauaro o -3 ko 3.

$$b=\frac{3±45}{2\times 9}$$

Whakareatia 2 ki te 9.

$$b=\frac{3±45}{18}$$

Nā, me whakaoti te whārite b=\frac{3±45}{18} ina he tāpiri te ±. Tāpiri 3 ki te 45.

$$b=\frac{48}{18}$$

Whakahekea te hautanga \frac{48}{18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$b=\frac{8}{3}$$

Nā, me whakaoti te whārite b=\frac{3±45}{18} ina he tango te ±. Tango 45 mai i 3.

$$b=-\frac{42}{18}$$

Whakahekea te hautanga \frac{-42}{18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$b=-\frac{7}{3}$$

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{8}{3} mō te x_{1} me te -\frac{7}{3} mō te x_{2}.

$$9b^{2}-3b-56=9\left(b-\frac{8}{3}\right)\left(b-\left(-\frac{7}{3}\right)\right)$$

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

$$9b^{2}-3b-56=9\left(b-\frac{8}{3}\right)\left(b+\frac{7}{3}\right)$$

Tango \frac{8}{3} 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.

$$9b^{2}-3b-56=9\times \frac{3b-8}{3}\left(b+\frac{7}{3}\right)$$

Tāpiri \frac{7}{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.

$$9b^{2}-3b-56=9\times \frac{3b-8}{3}\times \frac{3b+7}{3}$$

Whakareatia \frac{3b-8}{3} ki te \frac{3b+7}{3} 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.

$$9b^{2}-3b-56=9\times \frac{\left(3b-8\right)\left(3b+7\right)}{3\times 3}$$

Whakareatia 3 ki te 3.

$$9b^{2}-3b-56=9\times \frac{\left(3b-8\right)\left(3b+7\right)}{9}$$

Whakakorea atu te tauwehe pūnoa nui rawa 9 i roto i te 9 me te 9.

$$9b^{2}-3b-56=\left(3b-8\right)\left(3b+7\right)$$

Aromātai

$$\left(3b-8\right)\left(3b+7\right)$$