Whakaoti i te 21+m-10m^2

$$21 + m - 10 m ^ { 2 }$$

Tauwehe

$$\left(-5m-7\right)\left(2m-3\right)$$

Ngā Upane Otinga

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$-10m^{2}+m+21$$

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -10m^{2}+am+bm+21. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=1 ab=-10\times 21=-210$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -210.

$$-1,210 -2,105 -3,70 -5,42 -6,35 -7,30 -10,21 -14,15$$

Tātaihia te tapeke mō ia takirua.

$$-1+210=209 -2+105=103 -3+70=67 -5+42=37 -6+35=29 -7+30=23 -10+21=11 -14+15=1$$

Ko te otinga te takirua ka hoatu i te tapeke 1.

$$a=15 b=-14$$

Tuhia anō te -10m^{2}+m+21 hei \left(-10m^{2}+15m\right)+\left(-14m+21\right).

$$\left(-10m^{2}+15m\right)+\left(-14m+21\right)$$

Tauwehea te -5m i te tuatahi me te -7 i te rōpū tuarua.

$$-5m\left(2m-3\right)-7\left(2m-3\right)$$

Whakatauwehea atu te kīanga pātahi 2m-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2m-3\right)\left(-5m-7\right)$$

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.

$$-10m^{2}+m+21=0$$

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.

$$m=\frac{-1±\sqrt{1^{2}-4\left(-10\right)\times 21}}{2\left(-10\right)}$$

Pūrua 1.

$$m=\frac{-1±\sqrt{1-4\left(-10\right)\times 21}}{2\left(-10\right)}$$

Whakareatia -4 ki te -10.

$$m=\frac{-1±\sqrt{1+40\times 21}}{2\left(-10\right)}$$

Whakareatia 40 ki te 21.

$$m=\frac{-1±\sqrt{1+840}}{2\left(-10\right)}$$

Tāpiri 1 ki te 840.

$$m=\frac{-1±\sqrt{841}}{2\left(-10\right)}$$

Tuhia te pūtakerua o te 841.

$$m=\frac{-1±29}{2\left(-10\right)}$$

Whakareatia 2 ki te -10.

$$m=\frac{-1±29}{-20}$$

Nā, me whakaoti te whārite m=\frac{-1±29}{-20} ina he tāpiri te ±. Tāpiri -1 ki te 29.

$$m=\frac{28}{-20}$$

Whakahekea te hautanga \frac{28}{-20} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$m=-\frac{7}{5}$$

Nā, me whakaoti te whārite m=\frac{-1±29}{-20} ina he tango te ±. Tango 29 mai i -1.

$$m=-\frac{30}{-20}$$

Whakahekea te hautanga \frac{-30}{-20} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$m=\frac{3}{2}$$

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 -\frac{7}{5} mō te x_{1} me te \frac{3}{2} mō te x_{2}.

$$-10m^{2}+m+21=-10\left(m-\left(-\frac{7}{5}\right)\right)\left(m-\frac{3}{2}\right)$$

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

$$-10m^{2}+m+21=-10\left(m+\frac{7}{5}\right)\left(m-\frac{3}{2}\right)$$

Tāpiri \frac{7}{5} 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.

$$-10m^{2}+m+21=-10\times \frac{-5m-7}{-5}\left(m-\frac{3}{2}\right)$$

Tango \frac{3}{2} 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.

$$-10m^{2}+m+21=-10\times \frac{-5m-7}{-5}\times \frac{-2m+3}{-2}$$

Whakareatia \frac{-5m-7}{-5} ki te \frac{-2m+3}{-2} 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.

$$-10m^{2}+m+21=-10\times \frac{\left(-5m-7\right)\left(-2m+3\right)}{-5\left(-2\right)}$$

Whakareatia -5 ki te -2.

$$-10m^{2}+m+21=-10\times \frac{\left(-5m-7\right)\left(-2m+3\right)}{10}$$

Whakakorea atu te tauwehe pūnoa nui rawa 10 i roto i te -10 me te 10.

$$-10m^{2}+m+21=-\left(-5m-7\right)\left(-2m+3\right)$$

Aromātai

$$21+m-10m^{2}$$