Whakaoti i te 3q^2-135q+1350

$$3 q ^ { 2 } - 135 q + 1350$$

Tauwehe

$$3\left(q-30\right)\left(q-15\right)$$

Ngā Upane Otinga

Tauwehea te 3.

$$3\left(q^{2}-45q+450\right)$$

Whakaarohia te q^{2}-45q+450. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei q^{2}+aq+bq+450. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-45 ab=1\times 450=450$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 450.

$$-1,-450 -2,-225 -3,-150 -5,-90 -6,-75 -9,-50 -10,-45 -15,-30 -18,-25$$

Tātaihia te tapeke mō ia takirua.

$$-1-450=-451 -2-225=-227 -3-150=-153 -5-90=-95 -6-75=-81 -9-50=-59 -10-45=-55 -15-30=-45 -18-25=-43$$

Ko te otinga te takirua ka hoatu i te tapeke -45.

$$a=-30 b=-15$$

Tuhia anō te q^{2}-45q+450 hei \left(q^{2}-30q\right)+\left(-15q+450\right).

$$\left(q^{2}-30q\right)+\left(-15q+450\right)$$

Tauwehea te q i te tuatahi me te -15 i te rōpū tuarua.

$$q\left(q-30\right)-15\left(q-30\right)$$

Whakatauwehea atu te kīanga pātahi q-30 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(q-30\right)\left(q-15\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$3\left(q-30\right)\left(q-15\right)$$

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.

$$3q^{2}-135q+1350=0$$

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.

$$q=\frac{-\left(-135\right)±\sqrt{\left(-135\right)^{2}-4\times 3\times 1350}}{2\times 3}$$

Pūrua -135.

$$q=\frac{-\left(-135\right)±\sqrt{18225-4\times 3\times 1350}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$q=\frac{-\left(-135\right)±\sqrt{18225-12\times 1350}}{2\times 3}$$

Whakareatia -12 ki te 1350.

$$q=\frac{-\left(-135\right)±\sqrt{18225-16200}}{2\times 3}$$

Tāpiri 18225 ki te -16200.

$$q=\frac{-\left(-135\right)±\sqrt{2025}}{2\times 3}$$

Tuhia te pūtakerua o te 2025.

$$q=\frac{-\left(-135\right)±45}{2\times 3}$$

Ko te tauaro o -135 ko 135.

$$q=\frac{135±45}{2\times 3}$$

Whakareatia 2 ki te 3.

$$q=\frac{135±45}{6}$$

Nā, me whakaoti te whārite q=\frac{135±45}{6} ina he tāpiri te ±. Tāpiri 135 ki te 45.

$$q=\frac{180}{6}$$

Whakawehe 180 ki te 6.

$$q=30$$

Nā, me whakaoti te whārite q=\frac{135±45}{6} ina he tango te ±. Tango 45 mai i 135.

$$q=\frac{90}{6}$$

Whakawehe 90 ki te 6.

$$q=15$$

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 30 mō te x_{1} me te 15 mō te x_{2}.

$$3q^{2}-135q+1350=3\left(q-30\right)\left(q-15\right)$$

Aromātai

$$3\left(q-30\right)\left(q-15\right)$$