Whakaoti i te 3q^2-143q+1602

$$3 q ^ { 2 } - 143 q + 1602$$

Tauwehe

$$\left(q-18\right)\left(3q-89\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 3q^{2}+aq+bq+1602. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-143 ab=3\times 1602=4806$$

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 4806.

$$-1,-4806 -2,-2403 -3,-1602 -6,-801 -9,-534 -18,-267 -27,-178 -54,-89$$

Tātaihia te tapeke mō ia takirua.

$$-1-4806=-4807 -2-2403=-2405 -3-1602=-1605 -6-801=-807 -9-534=-543 -18-267=-285 -27-178=-205 -54-89=-143$$

Ko te otinga te takirua ka hoatu i te tapeke -143.

$$a=-89 b=-54$$

Tuhia anō te 3q^{2}-143q+1602 hei \left(3q^{2}-89q\right)+\left(-54q+1602\right).

$$\left(3q^{2}-89q\right)+\left(-54q+1602\right)$$

Tauwehea te q i te tuatahi me te -18 i te rōpū tuarua.

$$q\left(3q-89\right)-18\left(3q-89\right)$$

Whakatauwehea atu te kīanga pātahi 3q-89 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3q-89\right)\left(q-18\right)$$

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.

$$3q^{2}-143q+1602=0$$

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.

$$q=\frac{-\left(-143\right)±\sqrt{\left(-143\right)^{2}-4\times 3\times 1602}}{2\times 3}$$

Pūrua -143.

$$q=\frac{-\left(-143\right)±\sqrt{20449-4\times 3\times 1602}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$q=\frac{-\left(-143\right)±\sqrt{20449-12\times 1602}}{2\times 3}$$

Whakareatia -12 ki te 1602.

$$q=\frac{-\left(-143\right)±\sqrt{20449-19224}}{2\times 3}$$

Tāpiri 20449 ki te -19224.

$$q=\frac{-\left(-143\right)±\sqrt{1225}}{2\times 3}$$

Tuhia te pūtakerua o te 1225.

$$q=\frac{-\left(-143\right)±35}{2\times 3}$$

Ko te tauaro o -143 ko 143.

$$q=\frac{143±35}{2\times 3}$$

Whakareatia 2 ki te 3.

$$q=\frac{143±35}{6}$$

Nā, me whakaoti te whārite q=\frac{143±35}{6} ina he tāpiri te ±. Tāpiri 143 ki te 35.

$$q=\frac{178}{6}$$

Whakahekea te hautanga \frac{178}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$q=\frac{89}{3}$$

Nā, me whakaoti te whārite q=\frac{143±35}{6} ina he tango te ±. Tango 35 mai i 143.

$$q=\frac{108}{6}$$

Whakawehe 108 ki te 6.

$$q=18$$

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{89}{3} mō te x_{1} me te 18 mō te x_{2}.

$$3q^{2}-143q+1602=3\left(q-\frac{89}{3}\right)\left(q-18\right)$$

Tango \frac{89}{3} mai i q 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.

$$3q^{2}-143q+1602=3\times \frac{3q-89}{3}\left(q-18\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 3 i roto i te 3 me te 3.

$$3q^{2}-143q+1602=\left(3q-89\right)\left(q-18\right)$$

Aromātai

$$\left(q-18\right)\left(3q-89\right)$$