Whakaoti i te 4y^2-24y+27

$$4 y ^ { 2 } - 24 y + 27$$

Tauwehe

$$\left(2y-9\right)\left(2y-3\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 4y^{2}+ay+by+27. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-24 ab=4\times 27=108$$

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 108.

$$-1,-108 -2,-54 -3,-36 -4,-27 -6,-18 -9,-12$$

Tātaihia te tapeke mō ia takirua.

$$-1-108=-109 -2-54=-56 -3-36=-39 -4-27=-31 -6-18=-24 -9-12=-21$$

Ko te otinga te takirua ka hoatu i te tapeke -24.

$$a=-18 b=-6$$

Tuhia anō te 4y^{2}-24y+27 hei \left(4y^{2}-18y\right)+\left(-6y+27\right).

$$\left(4y^{2}-18y\right)+\left(-6y+27\right)$$

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

$$2y\left(2y-9\right)-3\left(2y-9\right)$$

Whakatauwehea atu te kīanga pātahi 2y-9 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2y-9\right)\left(2y-3\right)$$

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.

$$4y^{2}-24y+27=0$$

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.

$$y=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 4\times 27}}{2\times 4}$$

Pūrua -24.

$$y=\frac{-\left(-24\right)±\sqrt{576-4\times 4\times 27}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$y=\frac{-\left(-24\right)±\sqrt{576-16\times 27}}{2\times 4}$$

Whakareatia -16 ki te 27.

$$y=\frac{-\left(-24\right)±\sqrt{576-432}}{2\times 4}$$

Tāpiri 576 ki te -432.

$$y=\frac{-\left(-24\right)±\sqrt{144}}{2\times 4}$$

Tuhia te pūtakerua o te 144.

$$y=\frac{-\left(-24\right)±12}{2\times 4}$$

Ko te tauaro o -24 ko 24.

$$y=\frac{24±12}{2\times 4}$$

Whakareatia 2 ki te 4.

$$y=\frac{24±12}{8}$$

Nā, me whakaoti te whārite y=\frac{24±12}{8} ina he tāpiri te ±. Tāpiri 24 ki te 12.

$$y=\frac{36}{8}$$

Whakahekea te hautanga \frac{36}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$y=\frac{9}{2}$$

Nā, me whakaoti te whārite y=\frac{24±12}{8} ina he tango te ±. Tango 12 mai i 24.

$$y=\frac{12}{8}$$

Whakahekea te hautanga \frac{12}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

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

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{9}{2} mō te x_{1} me te \frac{3}{2} mō te x_{2}.

$$4y^{2}-24y+27=4\left(y-\frac{9}{2}\right)\left(y-\frac{3}{2}\right)$$

Tango \frac{9}{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.

$$4y^{2}-24y+27=4\times \frac{2y-9}{2}\left(y-\frac{3}{2}\right)$$

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.

$$4y^{2}-24y+27=4\times \frac{2y-9}{2}\times \frac{2y-3}{2}$$

Whakareatia \frac{2y-9}{2} 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.

$$4y^{2}-24y+27=4\times \frac{\left(2y-9\right)\left(2y-3\right)}{2\times 2}$$

Whakareatia 2 ki te 2.

$$4y^{2}-24y+27=4\times \frac{\left(2y-9\right)\left(2y-3\right)}{4}$$

Whakakorea atu te tauwehe pūnoa nui rawa 4 i roto i te 4 me te 4.

$$4y^{2}-24y+27=\left(2y-9\right)\left(2y-3\right)$$

Aromātai

$$\left(2y-9\right)\left(2y-3\right)$$