Whakaoti i te 3b^2-22b-80

$$3 b ^ { 2 } - 22 b - 80$$

Tauwehe

$$\left(b-10\right)\left(3b+8\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 3b^{2}+pb+qb-80. Hei kimi p me q, whakaritea tētahi pūnaha kia whakaoti.

$$p+q=-22 pq=3\left(-80\right)=-240$$

I te mea kua tōraro te pq, he tauaro ngā tohu o p me q. I te mea kua tōraro te p+q, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -240.

$$1,-240 2,-120 3,-80 4,-60 5,-48 6,-40 8,-30 10,-24 12,-20 15,-16$$

Tātaihia te tapeke mō ia takirua.

$$1-240=-239 2-120=-118 3-80=-77 4-60=-56 5-48=-43 6-40=-34 8-30=-22 10-24=-14 12-20=-8 15-16=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -22.

$$p=-30 q=8$$

Tuhia anō te 3b^{2}-22b-80 hei \left(3b^{2}-30b\right)+\left(8b-80\right).

$$\left(3b^{2}-30b\right)+\left(8b-80\right)$$

Tauwehea te 3b i te tuatahi me te 8 i te rōpū tuarua.

$$3b\left(b-10\right)+8\left(b-10\right)$$

Whakatauwehea atu te kīanga pātahi b-10 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(b-10\right)\left(3b+8\right)$$

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.

$$3b^{2}-22b-80=0$$

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.

$$b=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 3\left(-80\right)}}{2\times 3}$$

Pūrua -22.

$$b=\frac{-\left(-22\right)±\sqrt{484-4\times 3\left(-80\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$b=\frac{-\left(-22\right)±\sqrt{484-12\left(-80\right)}}{2\times 3}$$

Whakareatia -12 ki te -80.

$$b=\frac{-\left(-22\right)±\sqrt{484+960}}{2\times 3}$$

Tāpiri 484 ki te 960.

$$b=\frac{-\left(-22\right)±\sqrt{1444}}{2\times 3}$$

Tuhia te pūtakerua o te 1444.

$$b=\frac{-\left(-22\right)±38}{2\times 3}$$

Ko te tauaro o -22 ko 22.

$$b=\frac{22±38}{2\times 3}$$

Whakareatia 2 ki te 3.

$$b=\frac{22±38}{6}$$

Nā, me whakaoti te whārite b=\frac{22±38}{6} ina he tāpiri te ±. Tāpiri 22 ki te 38.

$$b=\frac{60}{6}$$

Whakawehe 60 ki te 6.

$$b=10$$

Nā, me whakaoti te whārite b=\frac{22±38}{6} ina he tango te ±. Tango 38 mai i 22.

$$b=-\frac{16}{6}$$

Whakahekea te hautanga \frac{-16}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$b=-\frac{8}{3}$$

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 10 mō te x_{1} me te -\frac{8}{3} mō te x_{2}.

$$3b^{2}-22b-80=3\left(b-10\right)\left(b-\left(-\frac{8}{3}\right)\right)$$

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

$$3b^{2}-22b-80=3\left(b-10\right)\left(b+\frac{8}{3}\right)$$

Tāpiri \frac{8}{3} ki te b 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.

$$3b^{2}-22b-80=3\left(b-10\right)\times \frac{3b+8}{3}$$

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

$$3b^{2}-22b-80=\left(b-10\right)\left(3b+8\right)$$

Aromātai

$$\left(b-10\right)\left(3b+8\right)$$