Whakaoti i te 12-7x-10x^2

$$12 - 7 x - 10 x ^ { 2 }$$

Tauwehe

$$\left(4-5x\right)\left(2x+3\right)$$

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.

$$-10x^{2}-7x+12$$

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -10x^{2}+ax+bx+12. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-7 ab=-10\times 12=-120$$

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 -120.

$$1,-120 2,-60 3,-40 4,-30 5,-24 6,-20 8,-15 10,-12$$

Tātaihia te tapeke mō ia takirua.

$$1-120=-119 2-60=-58 3-40=-37 4-30=-26 5-24=-19 6-20=-14 8-15=-7 10-12=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -7.

$$a=8 b=-15$$

Tuhia anō te -10x^{2}-7x+12 hei \left(-10x^{2}+8x\right)+\left(-15x+12\right).

$$\left(-10x^{2}+8x\right)+\left(-15x+12\right)$$

Tauwehea te 2x i te tuatahi me te 3 i te rōpū tuarua.

$$2x\left(-5x+4\right)+3\left(-5x+4\right)$$

Whakatauwehea atu te kīanga pātahi -5x+4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-5x+4\right)\left(2x+3\right)$$

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.

$$-10x^{2}-7x+12=0$$

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.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-10\right)\times 12}}{2\left(-10\right)}$$

Pūrua -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-10\right)\times 12}}{2\left(-10\right)}$$

Whakareatia -4 ki te -10.

$$x=\frac{-\left(-7\right)±\sqrt{49+40\times 12}}{2\left(-10\right)}$$

Whakareatia 40 ki te 12.

$$x=\frac{-\left(-7\right)±\sqrt{49+480}}{2\left(-10\right)}$$

Tāpiri 49 ki te 480.

$$x=\frac{-\left(-7\right)±\sqrt{529}}{2\left(-10\right)}$$

Tuhia te pūtakerua o te 529.

$$x=\frac{-\left(-7\right)±23}{2\left(-10\right)}$$

Ko te tauaro o -7 ko 7.

$$x=\frac{7±23}{2\left(-10\right)}$$

Whakareatia 2 ki te -10.

$$x=\frac{7±23}{-20}$$

Nā, me whakaoti te whārite x=\frac{7±23}{-20} ina he tāpiri te ±. Tāpiri 7 ki te 23.

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

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

$$x=-\frac{3}{2}$$

Nā, me whakaoti te whārite x=\frac{7±23}{-20} ina he tango te ±. Tango 23 mai i 7.

$$x=-\frac{16}{-20}$$

Whakahekea te hautanga \frac{-16}{-20} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{4}{5}$$

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{3}{2} mō te x_{1} me te \frac{4}{5} mō te x_{2}.

$$-10x^{2}-7x+12=-10\left(x-\left(-\frac{3}{2}\right)\right)\left(x-\frac{4}{5}\right)$$

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

$$-10x^{2}-7x+12=-10\left(x+\frac{3}{2}\right)\left(x-\frac{4}{5}\right)$$

Tāpiri \frac{3}{2} ki te x 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.

$$-10x^{2}-7x+12=-10\times \frac{-2x-3}{-2}\left(x-\frac{4}{5}\right)$$

Tango \frac{4}{5} mai i x 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.

$$-10x^{2}-7x+12=-10\times \frac{-2x-3}{-2}\times \frac{-5x+4}{-5}$$

Whakareatia \frac{-2x-3}{-2} ki te \frac{-5x+4}{-5} 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.

$$-10x^{2}-7x+12=-10\times \frac{\left(-2x-3\right)\left(-5x+4\right)}{-2\left(-5\right)}$$

Whakareatia -2 ki te -5.

$$-10x^{2}-7x+12=-10\times \frac{\left(-2x-3\right)\left(-5x+4\right)}{10}$$

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

$$-10x^{2}-7x+12=-\left(-2x-3\right)\left(-5x+4\right)$$

Aromātai

$$\left(4-5x\right)\left(2x+3\right)$$