Whakaoti i te 42m^2-89m-21

$$42 m ^ { 2 } - 89 m - 21$$

Tauwehe

$$\left(3m-7\right)\left(14m+3\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 42m^{2}+am+bm-21. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-89 ab=42\left(-21\right)=-882$$

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 -882.

$$1,-882 2,-441 3,-294 6,-147 7,-126 9,-98 14,-63 18,-49 21,-42$$

Tātaihia te tapeke mō ia takirua.

$$1-882=-881 2-441=-439 3-294=-291 6-147=-141 7-126=-119 9-98=-89 14-63=-49 18-49=-31 21-42=-21$$

Ko te otinga te takirua ka hoatu i te tapeke -89.

$$a=-98 b=9$$

Tuhia anō te 42m^{2}-89m-21 hei \left(42m^{2}-98m\right)+\left(9m-21\right).

$$\left(42m^{2}-98m\right)+\left(9m-21\right)$$

Tauwehea te 14m i te tuatahi me te 3 i te rōpū tuarua.

$$14m\left(3m-7\right)+3\left(3m-7\right)$$

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

$$\left(3m-7\right)\left(14m+3\right)$$

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.

$$42m^{2}-89m-21=0$$

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.

$$m=\frac{-\left(-89\right)±\sqrt{\left(-89\right)^{2}-4\times 42\left(-21\right)}}{2\times 42}$$

Pūrua -89.

$$m=\frac{-\left(-89\right)±\sqrt{7921-4\times 42\left(-21\right)}}{2\times 42}$$

Whakareatia -4 ki te 42.

$$m=\frac{-\left(-89\right)±\sqrt{7921-168\left(-21\right)}}{2\times 42}$$

Whakareatia -168 ki te -21.

$$m=\frac{-\left(-89\right)±\sqrt{7921+3528}}{2\times 42}$$

Tāpiri 7921 ki te 3528.

$$m=\frac{-\left(-89\right)±\sqrt{11449}}{2\times 42}$$

Tuhia te pūtakerua o te 11449.

$$m=\frac{-\left(-89\right)±107}{2\times 42}$$

Ko te tauaro o -89 ko 89.

$$m=\frac{89±107}{2\times 42}$$

Whakareatia 2 ki te 42.

$$m=\frac{89±107}{84}$$

Nā, me whakaoti te whārite m=\frac{89±107}{84} ina he tāpiri te ±. Tāpiri 89 ki te 107.

$$m=\frac{196}{84}$$

Whakahekea te hautanga \frac{196}{84} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 28.

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

Nā, me whakaoti te whārite m=\frac{89±107}{84} ina he tango te ±. Tango 107 mai i 89.

$$m=-\frac{18}{84}$$

Whakahekea te hautanga \frac{-18}{84} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$m=-\frac{3}{14}$$

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{3}{14} mō te x_{2}.

$$42m^{2}-89m-21=42\left(m-\frac{7}{3}\right)\left(m-\left(-\frac{3}{14}\right)\right)$$

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

$$42m^{2}-89m-21=42\left(m-\frac{7}{3}\right)\left(m+\frac{3}{14}\right)$$

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.

$$42m^{2}-89m-21=42\times \frac{3m-7}{3}\left(m+\frac{3}{14}\right)$$

Tāpiri \frac{3}{14} 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.

$$42m^{2}-89m-21=42\times \frac{3m-7}{3}\times \frac{14m+3}{14}$$

Whakareatia \frac{3m-7}{3} ki te \frac{14m+3}{14} 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.

$$42m^{2}-89m-21=42\times \frac{\left(3m-7\right)\left(14m+3\right)}{3\times 14}$$

Whakareatia 3 ki te 14.

$$42m^{2}-89m-21=42\times \frac{\left(3m-7\right)\left(14m+3\right)}{42}$$

Whakakorea atu te tauwehe pūnoa nui rawa 42 i roto i te 42 me te 42.

$$42m^{2}-89m-21=\left(3m-7\right)\left(14m+3\right)$$

Aromātai

$$\left(3m-7\right)\left(14m+3\right)$$