Whakaoti i te -16+72x-81x^2

$$- 16 + 72 x - 81 x ^ { 2 }$$

Tauwehe

$$-\left(9x-4\right)^{2}$$

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.

$$-81x^{2}+72x-16$$

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -81x^{2}+ax+bx-16. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=72 ab=-81\left(-16\right)=1296$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 1296.

$$1,1296 2,648 3,432 4,324 6,216 8,162 9,144 12,108 16,81 18,72 24,54 27,48 36,36$$

Tātaihia te tapeke mō ia takirua.

$$1+1296=1297 2+648=650 3+432=435 4+324=328 6+216=222 8+162=170 9+144=153 12+108=120 16+81=97 18+72=90 24+54=78 27+48=75 36+36=72$$

Ko te otinga te takirua ka hoatu i te tapeke 72.

$$a=36 b=36$$

Tuhia anō te -81x^{2}+72x-16 hei \left(-81x^{2}+36x\right)+\left(36x-16\right).

$$\left(-81x^{2}+36x\right)+\left(36x-16\right)$$

Tauwehea te -9x i te tuatahi me te 4 i te rōpū tuarua.

$$-9x\left(9x-4\right)+4\left(9x-4\right)$$

Whakatauwehea atu te kīanga pātahi 9x-4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(9x-4\right)\left(-9x+4\right)$$

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.

$$-81x^{2}+72x-16=0$$

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.

$$x=\frac{-72±\sqrt{72^{2}-4\left(-81\right)\left(-16\right)}}{2\left(-81\right)}$$

Pūrua 72.

$$x=\frac{-72±\sqrt{5184-4\left(-81\right)\left(-16\right)}}{2\left(-81\right)}$$

Whakareatia -4 ki te -81.

$$x=\frac{-72±\sqrt{5184+324\left(-16\right)}}{2\left(-81\right)}$$

Whakareatia 324 ki te -16.

$$x=\frac{-72±\sqrt{5184-5184}}{2\left(-81\right)}$$

Tāpiri 5184 ki te -5184.

$$x=\frac{-72±\sqrt{0}}{2\left(-81\right)}$$

Tuhia te pūtakerua o te 0.

$$x=\frac{-72±0}{2\left(-81\right)}$$

Whakareatia 2 ki te -81.

$$x=\frac{-72±0}{-162}$$

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{4}{9} mō te x_{1} me te \frac{4}{9} mō te x_{2}.

$$-81x^{2}+72x-16=-81\left(x-\frac{4}{9}\right)\left(x-\frac{4}{9}\right)$$

Tango \frac{4}{9} 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.

$$-81x^{2}+72x-16=-81\times \frac{-9x+4}{-9}\left(x-\frac{4}{9}\right)$$

Tango \frac{4}{9} 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.

$$-81x^{2}+72x-16=-81\times \frac{-9x+4}{-9}\times \frac{-9x+4}{-9}$$

Whakareatia \frac{-9x+4}{-9} ki te \frac{-9x+4}{-9} 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.

$$-81x^{2}+72x-16=-81\times \frac{\left(-9x+4\right)\left(-9x+4\right)}{-9\left(-9\right)}$$

Whakareatia -9 ki te -9.

$$-81x^{2}+72x-16=-81\times \frac{\left(-9x+4\right)\left(-9x+4\right)}{81}$$

Whakakorea atu te tauwehe pūnoa nui rawa 81 i roto i te -81 me te 81.

$$-81x^{2}+72x-16=-\left(-9x+4\right)\left(-9x+4\right)$$

Aromātai

$$-\left(9x-4\right)^{2}$$