Whakaoti i te 2y^2-9y-18

$$2 y ^ { 2 } - 9 y - 18$$

Tauwehe

$$\left(y-6\right)\left(2y+3\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 2y^{2}+ay+by-18. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-9 ab=2\left(-18\right)=-36$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, 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 -36.

$$1,-36 2,-18 3,-12 4,-9 6,-6$$

Tātaihia te tapeke mō ia takirua.

$$1-36=-35 2-18=-16 3-12=-9 4-9=-5 6-6=0$$

Ko te otinga te takirua ka hoatu i te tapeke -9.

$$a=-12 b=3$$

Tuhia anō te 2y^{2}-9y-18 hei \left(2y^{2}-12y\right)+\left(3y-18\right).

$$\left(2y^{2}-12y\right)+\left(3y-18\right)$$

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

$$2y\left(y-6\right)+3\left(y-6\right)$$

Whakatauwehea atu te kīanga pātahi y-6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(y-6\right)\left(2y+3\right)$$

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.

$$2y^{2}-9y-18=0$$

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.

$$y=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 2\left(-18\right)}}{2\times 2}$$

Pūrua -9.

$$y=\frac{-\left(-9\right)±\sqrt{81-4\times 2\left(-18\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$y=\frac{-\left(-9\right)±\sqrt{81-8\left(-18\right)}}{2\times 2}$$

Whakareatia -8 ki te -18.

$$y=\frac{-\left(-9\right)±\sqrt{81+144}}{2\times 2}$$

Tāpiri 81 ki te 144.

$$y=\frac{-\left(-9\right)±\sqrt{225}}{2\times 2}$$

Tuhia te pūtakerua o te 225.

$$y=\frac{-\left(-9\right)±15}{2\times 2}$$

Ko te tauaro o -9 ko 9.

$$y=\frac{9±15}{2\times 2}$$

Whakareatia 2 ki te 2.

$$y=\frac{9±15}{4}$$

Nā, me whakaoti te whārite y=\frac{9±15}{4} ina he tāpiri te ±. Tāpiri 9 ki te 15.

$$y=\frac{24}{4}$$

Whakawehe 24 ki te 4.

$$y=6$$

Nā, me whakaoti te whārite y=\frac{9±15}{4} ina he tango te ±. Tango 15 mai i 9.

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

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

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

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

$$2y^{2}-9y-18=2\left(y-6\right)\left(y-\left(-\frac{3}{2}\right)\right)$$

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

$$2y^{2}-9y-18=2\left(y-6\right)\left(y+\frac{3}{2}\right)$$

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.

$$2y^{2}-9y-18=2\left(y-6\right)\times \frac{2y+3}{2}$$

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

$$2y^{2}-9y-18=\left(y-6\right)\left(2y+3\right)$$

Aromātai

$$\left(y-6\right)\left(2y+3\right)$$