Whakaoti i te 90m^2-137m-45

$$90 m ^ { 2 } - 137 m - 45$$

Tauwehe

$$\left(5m-9\right)\left(18m+5\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 90m^{2}+am+bm-45. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-137 ab=90\left(-45\right)=-4050$$

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 -4050.

$$1,-4050 2,-2025 3,-1350 5,-810 6,-675 9,-450 10,-405 15,-270 18,-225 25,-162 27,-150 30,-135 45,-90 50,-81 54,-75$$

Tātaihia te tapeke mō ia takirua.

$$1-4050=-4049 2-2025=-2023 3-1350=-1347 5-810=-805 6-675=-669 9-450=-441 10-405=-395 15-270=-255 18-225=-207 25-162=-137 27-150=-123 30-135=-105 45-90=-45 50-81=-31 54-75=-21$$

Ko te otinga te takirua ka hoatu i te tapeke -137.

$$a=-162 b=25$$

Tuhia anō te 90m^{2}-137m-45 hei \left(90m^{2}-162m\right)+\left(25m-45\right).

$$\left(90m^{2}-162m\right)+\left(25m-45\right)$$

Tauwehea te 18m i te tuatahi me te 5 i te rōpū tuarua.

$$18m\left(5m-9\right)+5\left(5m-9\right)$$

Whakatauwehea atu te kīanga pātahi 5m-9 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(5m-9\right)\left(18m+5\right)$$

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.

$$90m^{2}-137m-45=0$$

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.

$$m=\frac{-\left(-137\right)±\sqrt{\left(-137\right)^{2}-4\times 90\left(-45\right)}}{2\times 90}$$

Pūrua -137.

$$m=\frac{-\left(-137\right)±\sqrt{18769-4\times 90\left(-45\right)}}{2\times 90}$$

Whakareatia -4 ki te 90.

$$m=\frac{-\left(-137\right)±\sqrt{18769-360\left(-45\right)}}{2\times 90}$$

Whakareatia -360 ki te -45.

$$m=\frac{-\left(-137\right)±\sqrt{18769+16200}}{2\times 90}$$

Tāpiri 18769 ki te 16200.

$$m=\frac{-\left(-137\right)±\sqrt{34969}}{2\times 90}$$

Tuhia te pūtakerua o te 34969.

$$m=\frac{-\left(-137\right)±187}{2\times 90}$$

Ko te tauaro o -137 ko 137.

$$m=\frac{137±187}{2\times 90}$$

Whakareatia 2 ki te 90.

$$m=\frac{137±187}{180}$$

Nā, me whakaoti te whārite m=\frac{137±187}{180} ina he tāpiri te ±. Tāpiri 137 ki te 187.

$$m=\frac{324}{180}$$

Whakahekea te hautanga \frac{324}{180} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 36.

$$m=\frac{9}{5}$$

Nā, me whakaoti te whārite m=\frac{137±187}{180} ina he tango te ±. Tango 187 mai i 137.

$$m=-\frac{50}{180}$$

Whakahekea te hautanga \frac{-50}{180} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$m=-\frac{5}{18}$$

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{9}{5} mō te x_{1} me te -\frac{5}{18} mō te x_{2}.

$$90m^{2}-137m-45=90\left(m-\frac{9}{5}\right)\left(m-\left(-\frac{5}{18}\right)\right)$$

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

$$90m^{2}-137m-45=90\left(m-\frac{9}{5}\right)\left(m+\frac{5}{18}\right)$$

Tango \frac{9}{5} 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.

$$90m^{2}-137m-45=90\times \frac{5m-9}{5}\left(m+\frac{5}{18}\right)$$

Tāpiri \frac{5}{18} 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.

$$90m^{2}-137m-45=90\times \frac{5m-9}{5}\times \frac{18m+5}{18}$$

Whakareatia \frac{5m-9}{5} ki te \frac{18m+5}{18} 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.

$$90m^{2}-137m-45=90\times \frac{\left(5m-9\right)\left(18m+5\right)}{5\times 18}$$

Whakareatia 5 ki te 18.

$$90m^{2}-137m-45=90\times \frac{\left(5m-9\right)\left(18m+5\right)}{90}$$

Whakakorea atu te tauwehe pūnoa nui rawa 90 i roto i te 90 me te 90.

$$90m^{2}-137m-45=\left(5m-9\right)\left(18m+5\right)$$

Aromātai

$$\left(5m-9\right)\left(18m+5\right)$$