Whakaoti i te 14y^2+37y+24

$$14 { y }^{ 2 } +37y+24$$

Tauwehe

$$\left(2y+3\right)\left(7y+8\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 14y^{2}+ay+by+24. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=37 ab=14\times 24=336$$

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 336.

$$1,336 2,168 3,112 4,84 6,56 7,48 8,42 12,28 14,24 16,21$$

Tātaihia te tapeke mō ia takirua.

$$1+336=337 2+168=170 3+112=115 4+84=88 6+56=62 7+48=55 8+42=50 12+28=40 14+24=38 16+21=37$$

Ko te otinga te takirua ka hoatu i te tapeke 37.

$$a=16 b=21$$

Tuhia anō te 14y^{2}+37y+24 hei \left(14y^{2}+16y\right)+\left(21y+24\right).

$$\left(14y^{2}+16y\right)+\left(21y+24\right)$$

Tauwehea te 2y i te tuatahi me te 3 i te rōpū tuarua.

$$2y\left(7y+8\right)+3\left(7y+8\right)$$

Whakatauwehea atu te kīanga pātahi 7y+8 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(7y+8\right)\left(2y+3\right)$$

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.

$$14y^{2}+37y+24=0$$

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.

$$y=\frac{-37±\sqrt{37^{2}-4\times 14\times 24}}{2\times 14}$$

Pūrua 37.

$$y=\frac{-37±\sqrt{1369-4\times 14\times 24}}{2\times 14}$$

Whakareatia -4 ki te 14.

$$y=\frac{-37±\sqrt{1369-56\times 24}}{2\times 14}$$

Whakareatia -56 ki te 24.

$$y=\frac{-37±\sqrt{1369-1344}}{2\times 14}$$

Tāpiri 1369 ki te -1344.

$$y=\frac{-37±\sqrt{25}}{2\times 14}$$

Tuhia te pūtakerua o te 25.

$$y=\frac{-37±5}{2\times 14}$$

Whakareatia 2 ki te 14.

$$y=\frac{-37±5}{28}$$

Nā, me whakaoti te whārite y=\frac{-37±5}{28} ina he tāpiri te ±. Tāpiri -37 ki te 5.

$$y=-\frac{32}{28}$$

Whakahekea te hautanga \frac{-32}{28} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$y=-\frac{8}{7}$$

Nā, me whakaoti te whārite y=\frac{-37±5}{28} ina he tango te ±. Tango 5 mai i -37.

$$y=-\frac{42}{28}$$

Whakahekea te hautanga \frac{-42}{28} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 14.

$$y=-\frac{3}{2}$$

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{8}{7} mō te x_{1} me te -\frac{3}{2} mō te x_{2}.

$$14y^{2}+37y+24=14\left(y-\left(-\frac{8}{7}\right)\right)\left(y-\left(-\frac{3}{2}\right)\right)$$

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

$$14y^{2}+37y+24=14\left(y+\frac{8}{7}\right)\left(y+\frac{3}{2}\right)$$

Tāpiri \frac{8}{7} ki te y 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.

$$14y^{2}+37y+24=14\times \frac{7y+8}{7}\left(y+\frac{3}{2}\right)$$

Tāpiri \frac{3}{2} ki te y 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.

$$14y^{2}+37y+24=14\times \frac{7y+8}{7}\times \frac{2y+3}{2}$$

Whakareatia \frac{7y+8}{7} ki te \frac{2y+3}{2} 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.

$$14y^{2}+37y+24=14\times \frac{\left(7y+8\right)\left(2y+3\right)}{7\times 2}$$

Whakareatia 7 ki te 2.

$$14y^{2}+37y+24=14\times \frac{\left(7y+8\right)\left(2y+3\right)}{14}$$

Whakakorea atu te tauwehe pūnoa nui rawa 14 i roto i te 14 me te 14.

$$14y^{2}+37y+24=\left(7y+8\right)\left(2y+3\right)$$

Aromātai

$$\left(2y+3\right)\left(7y+8\right)$$