Whakaoti i te 2y^2-9y+4

$$2 y ^ { 2 } - 9 y + 4$$

Tauwehe

$$\left(y-4\right)\left(2y-1\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 2y^{2}+ay+by+4. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-9 ab=2\times 4=8$$

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 8.

$$-1,-8 -2,-4$$

Tātaihia te tapeke mō ia takirua.

$$-1-8=-9 -2-4=-6$$

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

$$a=-8 b=-1$$

Tuhia anō te 2y^{2}-9y+4 hei \left(2y^{2}-8y\right)+\left(-y+4\right).

$$\left(2y^{2}-8y\right)+\left(-y+4\right)$$

Tauwehea te 2y i te tuatahi me te -1 i te rōpū tuarua.

$$2y\left(y-4\right)-\left(y-4\right)$$

Whakatauwehea atu te kīanga pātahi y-4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(y-4\right)\left(2y-1\right)$$

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.

$$2y^{2}-9y+4=0$$

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.

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

Pūrua -9.

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

Whakareatia -4 ki te 2.

$$y=\frac{-\left(-9\right)±\sqrt{81-8\times 4}}{2\times 2}$$

Whakareatia -8 ki te 4.

$$y=\frac{-\left(-9\right)±\sqrt{81-32}}{2\times 2}$$

Tāpiri 81 ki te -32.

$$y=\frac{-\left(-9\right)±\sqrt{49}}{2\times 2}$$

Tuhia te pūtakerua o te 49.

$$y=\frac{-\left(-9\right)±7}{2\times 2}$$

Ko te tauaro o -9 ko 9.

$$y=\frac{9±7}{2\times 2}$$

Whakareatia 2 ki te 2.

$$y=\frac{9±7}{4}$$

Nā, me whakaoti te whārite y=\frac{9±7}{4} ina he tāpiri te ±. Tāpiri 9 ki te 7.

$$y=\frac{16}{4}$$

Whakawehe 16 ki te 4.

$$y=4$$

Nā, me whakaoti te whārite y=\frac{9±7}{4} ina he tango te ±. Tango 7 mai i 9.

$$y=\frac{2}{4}$$

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

$$y=\frac{1}{2}$$

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 4 mō te x_{1} me te \frac{1}{2} mō te x_{2}.

$$2y^{2}-9y+4=2\left(y-4\right)\left(y-\frac{1}{2}\right)$$

Tango \frac{1}{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.

$$2y^{2}-9y+4=2\left(y-4\right)\times \frac{2y-1}{2}$$

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

$$2y^{2}-9y+4=\left(y-4\right)\left(2y-1\right)$$

Aromātai

$$\left(y-4\right)\left(2y-1\right)$$