Whakaoti i te 9w^2+9w-4

$$9 w ^ { 2 } + 9 w - 4$$

Tauwehe

$$\left(3w-1\right)\left(3w+4\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 9w^{2}+aw+bw-4. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=9 ab=9\left(-4\right)=-36$$

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 -36.

$$-1,36 -2,18 -3,12 -4,9 -6,6$$

Tātaihia te tapeke mō ia takirua.

$$-1+36=35 -2+18=16 -3+12=9 -4+9=5 -6+6=0$$

Ko te otinga te takirua ka hoatu i te tapeke 9.

$$a=-3 b=12$$

Tuhia anō te 9w^{2}+9w-4 hei \left(9w^{2}-3w\right)+\left(12w-4\right).

$$\left(9w^{2}-3w\right)+\left(12w-4\right)$$

Tauwehea te 3w i te tuatahi me te 4 i te rōpū tuarua.

$$3w\left(3w-1\right)+4\left(3w-1\right)$$

Whakatauwehea atu te kīanga pātahi 3w-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3w-1\right)\left(3w+4\right)$$

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.

$$9w^{2}+9w-4=0$$

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.

$$w=\frac{-9±\sqrt{9^{2}-4\times 9\left(-4\right)}}{2\times 9}$$

Pūrua 9.

$$w=\frac{-9±\sqrt{81-4\times 9\left(-4\right)}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$w=\frac{-9±\sqrt{81-36\left(-4\right)}}{2\times 9}$$

Whakareatia -36 ki te -4.

$$w=\frac{-9±\sqrt{81+144}}{2\times 9}$$

Tāpiri 81 ki te 144.

$$w=\frac{-9±\sqrt{225}}{2\times 9}$$

Tuhia te pūtakerua o te 225.

$$w=\frac{-9±15}{2\times 9}$$

Whakareatia 2 ki te 9.

$$w=\frac{-9±15}{18}$$

Nā, me whakaoti te whārite w=\frac{-9±15}{18} ina he tāpiri te ±. Tāpiri -9 ki te 15.

$$w=\frac{6}{18}$$

Whakahekea te hautanga \frac{6}{18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$w=\frac{1}{3}$$

Nā, me whakaoti te whārite w=\frac{-9±15}{18} ina he tango te ±. Tango 15 mai i -9.

$$w=-\frac{24}{18}$$

Whakahekea te hautanga \frac{-24}{18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$w=-\frac{4}{3}$$

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

$$9w^{2}+9w-4=9\left(w-\frac{1}{3}\right)\left(w-\left(-\frac{4}{3}\right)\right)$$

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

$$9w^{2}+9w-4=9\left(w-\frac{1}{3}\right)\left(w+\frac{4}{3}\right)$$

Tango \frac{1}{3} mai i w 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.

$$9w^{2}+9w-4=9\times \frac{3w-1}{3}\left(w+\frac{4}{3}\right)$$

Tāpiri \frac{4}{3} ki te w 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.

$$9w^{2}+9w-4=9\times \frac{3w-1}{3}\times \frac{3w+4}{3}$$

Whakareatia \frac{3w-1}{3} ki te \frac{3w+4}{3} 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.

$$9w^{2}+9w-4=9\times \frac{\left(3w-1\right)\left(3w+4\right)}{3\times 3}$$

Whakareatia 3 ki te 3.

$$9w^{2}+9w-4=9\times \frac{\left(3w-1\right)\left(3w+4\right)}{9}$$

Whakakorea atu te tauwehe pūnoa nui rawa 9 i roto i te 9 me te 9.

$$9w^{2}+9w-4=\left(3w-1\right)\left(3w+4\right)$$

Aromātai

$$\left(3w-1\right)\left(3w+4\right)$$