Whakaoti i te 2y^2+13y-24

$$2 y ^ { 2 } + 13 y - 24$$

Tauwehe

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

Ngā Upane Otinga

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

$$a+b=13 ab=2\left(-24\right)=-48$$

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 -48.

$$-1,48 -2,24 -3,16 -4,12 -6,8$$

Tātaihia te tapeke mō ia takirua.

$$-1+48=47 -2+24=22 -3+16=13 -4+12=8 -6+8=2$$

Ko te otinga te takirua ka hoatu i te tapeke 13.

$$a=-3 b=16$$

Tuhia anō te 2y^{2}+13y-24 hei \left(2y^{2}-3y\right)+\left(16y-24\right).

$$\left(2y^{2}-3y\right)+\left(16y-24\right)$$

Tauwehea te y i te tuatahi me te 8 i te rōpū tuarua.

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

Whakatauwehea atu te kīanga pātahi 2y-3 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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.

$$2y^{2}+13y-24=0$$

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.

$$y=\frac{-13±\sqrt{13^{2}-4\times 2\left(-24\right)}}{2\times 2}$$

Pūrua 13.

$$y=\frac{-13±\sqrt{169-4\times 2\left(-24\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$y=\frac{-13±\sqrt{169-8\left(-24\right)}}{2\times 2}$$

Whakareatia -8 ki te -24.

$$y=\frac{-13±\sqrt{169+192}}{2\times 2}$$

Tāpiri 169 ki te 192.

$$y=\frac{-13±\sqrt{361}}{2\times 2}$$

Tuhia te pūtakerua o te 361.

$$y=\frac{-13±19}{2\times 2}$$

Whakareatia 2 ki te 2.

$$y=\frac{-13±19}{4}$$

Nā, me whakaoti te whārite y=\frac{-13±19}{4} ina he tāpiri te ±. Tāpiri -13 ki te 19.

$$y=\frac{6}{4}$$

Whakahekea te hautanga \frac{6}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Nā, me whakaoti te whārite y=\frac{-13±19}{4} ina he tango te ±. Tango 19 mai i -13.

$$y=-\frac{32}{4}$$

Whakawehe -32 ki te 4.

$$y=-8$$

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

$$2y^{2}+13y-24=2\left(y-\frac{3}{2}\right)\left(y-\left(-8\right)\right)$$

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

$$2y^{2}+13y-24=2\left(y-\frac{3}{2}\right)\left(y+8\right)$$

Tango \frac{3}{2} mai i y 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.

$$2y^{2}+13y-24=2\times \frac{2y-3}{2}\left(y+8\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 2 i roto i te 2 me te 2.

$$2y^{2}+13y-24=\left(2y-3\right)\left(y+8\right)$$

Aromātai

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