Whakaoti i te 27+30x-25x^2

$$27 + 30 x - 25 x ^ { 2 }$$

Tauwehe

$$-\left(5x-9\right)\left(5x+3\right)$$

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.

$$-25x^{2}+30x+27$$

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -25x^{2}+ax+bx+27. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=30 ab=-25\times 27=-675$$

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 -675.

$$-1,675 -3,225 -5,135 -9,75 -15,45 -25,27$$

Tātaihia te tapeke mō ia takirua.

$$-1+675=674 -3+225=222 -5+135=130 -9+75=66 -15+45=30 -25+27=2$$

Ko te otinga te takirua ka hoatu i te tapeke 30.

$$a=45 b=-15$$

Tuhia anō te -25x^{2}+30x+27 hei \left(-25x^{2}+45x\right)+\left(-15x+27\right).

$$\left(-25x^{2}+45x\right)+\left(-15x+27\right)$$

Tauwehea te -5x i te tuatahi me te -3 i te rōpū tuarua.

$$-5x\left(5x-9\right)-3\left(5x-9\right)$$

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

$$\left(5x-9\right)\left(-5x-3\right)$$

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.

$$-25x^{2}+30x+27=0$$

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.

$$x=\frac{-30±\sqrt{30^{2}-4\left(-25\right)\times 27}}{2\left(-25\right)}$$

Pūrua 30.

$$x=\frac{-30±\sqrt{900-4\left(-25\right)\times 27}}{2\left(-25\right)}$$

Whakareatia -4 ki te -25.

$$x=\frac{-30±\sqrt{900+100\times 27}}{2\left(-25\right)}$$

Whakareatia 100 ki te 27.

$$x=\frac{-30±\sqrt{900+2700}}{2\left(-25\right)}$$

Tāpiri 900 ki te 2700.

$$x=\frac{-30±\sqrt{3600}}{2\left(-25\right)}$$

Tuhia te pūtakerua o te 3600.

$$x=\frac{-30±60}{2\left(-25\right)}$$

Whakareatia 2 ki te -25.

$$x=\frac{-30±60}{-50}$$

Nā, me whakaoti te whārite x=\frac{-30±60}{-50} ina he tāpiri te ±. Tāpiri -30 ki te 60.

$$x=\frac{30}{-50}$$

Whakahekea te hautanga \frac{30}{-50} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$x=-\frac{3}{5}$$

Nā, me whakaoti te whārite x=\frac{-30±60}{-50} ina he tango te ±. Tango 60 mai i -30.

$$x=-\frac{90}{-50}$$

Whakahekea te hautanga \frac{-90}{-50} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

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

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

$$-25x^{2}+30x+27=-25\left(x-\left(-\frac{3}{5}\right)\right)\left(x-\frac{9}{5}\right)$$

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

$$-25x^{2}+30x+27=-25\left(x+\frac{3}{5}\right)\left(x-\frac{9}{5}\right)$$

Tāpiri \frac{3}{5} 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.

$$-25x^{2}+30x+27=-25\times \frac{-5x-3}{-5}\left(x-\frac{9}{5}\right)$$

Tango \frac{9}{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.

$$-25x^{2}+30x+27=-25\times \frac{-5x-3}{-5}\times \frac{-5x+9}{-5}$$

Whakareatia \frac{-5x-3}{-5} ki te \frac{-5x+9}{-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.

$$-25x^{2}+30x+27=-25\times \frac{\left(-5x-3\right)\left(-5x+9\right)}{-5\left(-5\right)}$$

Whakareatia -5 ki te -5.

$$-25x^{2}+30x+27=-25\times \frac{\left(-5x-3\right)\left(-5x+9\right)}{25}$$

Whakakorea atu te tauwehe pūnoa nui rawa 25 i roto i te -25 me te 25.

$$-25x^{2}+30x+27=-\left(-5x-3\right)\left(-5x+9\right)$$

Aromātai

$$27+30x-25x^{2}$$