Whakaoti i te 9m^2-9m-28

$$9 m ^ { 2 } - 9 m - 28$$

Tauwehe

$$\left(3m-7\right)\left(3m+4\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 9m^{2}+am+bm-28. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-9 ab=9\left(-28\right)=-252$$

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 -252.

$$1,-252 2,-126 3,-84 4,-63 6,-42 7,-36 9,-28 12,-21 14,-18$$

Tātaihia te tapeke mō ia takirua.

$$1-252=-251 2-126=-124 3-84=-81 4-63=-59 6-42=-36 7-36=-29 9-28=-19 12-21=-9 14-18=-4$$

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

$$a=-21 b=12$$

Tuhia anō te 9m^{2}-9m-28 hei \left(9m^{2}-21m\right)+\left(12m-28\right).

$$\left(9m^{2}-21m\right)+\left(12m-28\right)$$

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

$$3m\left(3m-7\right)+4\left(3m-7\right)$$

Whakatauwehea atu te kīanga pātahi 3m-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3m-7\right)\left(3m+4\right)$$

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.

$$9m^{2}-9m-28=0$$

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.

$$m=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 9\left(-28\right)}}{2\times 9}$$

Pūrua -9.

$$m=\frac{-\left(-9\right)±\sqrt{81-4\times 9\left(-28\right)}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$m=\frac{-\left(-9\right)±\sqrt{81-36\left(-28\right)}}{2\times 9}$$

Whakareatia -36 ki te -28.

$$m=\frac{-\left(-9\right)±\sqrt{81+1008}}{2\times 9}$$

Tāpiri 81 ki te 1008.

$$m=\frac{-\left(-9\right)±\sqrt{1089}}{2\times 9}$$

Tuhia te pūtakerua o te 1089.

$$m=\frac{-\left(-9\right)±33}{2\times 9}$$

Ko te tauaro o -9 ko 9.

$$m=\frac{9±33}{2\times 9}$$

Whakareatia 2 ki te 9.

$$m=\frac{9±33}{18}$$

Nā, me whakaoti te whārite m=\frac{9±33}{18} ina he tāpiri te ±. Tāpiri 9 ki te 33.

$$m=\frac{42}{18}$$

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

$$m=\frac{7}{3}$$

Nā, me whakaoti te whārite m=\frac{9±33}{18} ina he tango te ±. Tango 33 mai i 9.

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

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

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

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{7}{3} mō te x_{1} me te -\frac{4}{3} mō te x_{2}.

$$9m^{2}-9m-28=9\left(m-\frac{7}{3}\right)\left(m-\left(-\frac{4}{3}\right)\right)$$

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

$$9m^{2}-9m-28=9\left(m-\frac{7}{3}\right)\left(m+\frac{4}{3}\right)$$

Tango \frac{7}{3} mai i m 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.

$$9m^{2}-9m-28=9\times \frac{3m-7}{3}\left(m+\frac{4}{3}\right)$$

Tāpiri \frac{4}{3} ki te m 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.

$$9m^{2}-9m-28=9\times \frac{3m-7}{3}\times \frac{3m+4}{3}$$

Whakareatia \frac{3m-7}{3} ki te \frac{3m+4}{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.

$$9m^{2}-9m-28=9\times \frac{\left(3m-7\right)\left(3m+4\right)}{3\times 3}$$

Whakareatia 3 ki te 3.

$$9m^{2}-9m-28=9\times \frac{\left(3m-7\right)\left(3m+4\right)}{9}$$

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

$$9m^{2}-9m-28=\left(3m-7\right)\left(3m+4\right)$$

Aromātai

$$\left(3m-7\right)\left(3m+4\right)$$