Whakaoti i te 4y^2+35y-9

$$4 y ^ { 2 } + 35 y - 9$$

Tauwehe

$$\left(4y-1\right)\left(y+9\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 4y^{2}+ay+by-9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=35 ab=4\left(-9\right)=-36$$

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 -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 35.

$$a=-1 b=36$$

Tuhia anō te 4y^{2}+35y-9 hei \left(4y^{2}-y\right)+\left(36y-9\right).

$$\left(4y^{2}-y\right)+\left(36y-9\right)$$

Tauwehea te y i te tuatahi me te 9 i te rōpū tuarua.

$$y\left(4y-1\right)+9\left(4y-1\right)$$

Whakatauwehea atu te kīanga pātahi 4y-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(4y-1\right)\left(y+9\right)$$

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.

$$4y^{2}+35y-9=0$$

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.

$$y=\frac{-35±\sqrt{35^{2}-4\times 4\left(-9\right)}}{2\times 4}$$

Pūrua 35.

$$y=\frac{-35±\sqrt{1225-4\times 4\left(-9\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$y=\frac{-35±\sqrt{1225-16\left(-9\right)}}{2\times 4}$$

Whakareatia -16 ki te -9.

$$y=\frac{-35±\sqrt{1225+144}}{2\times 4}$$

Tāpiri 1225 ki te 144.

$$y=\frac{-35±\sqrt{1369}}{2\times 4}$$

Tuhia te pūtakerua o te 1369.

$$y=\frac{-35±37}{2\times 4}$$

Whakareatia 2 ki te 4.

$$y=\frac{-35±37}{8}$$

Nā, me whakaoti te whārite y=\frac{-35±37}{8} ina he tāpiri te ±. Tāpiri -35 ki te 37.

$$y=\frac{2}{8}$$

Whakahekea te hautanga \frac{2}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$y=\frac{1}{4}$$

Nā, me whakaoti te whārite y=\frac{-35±37}{8} ina he tango te ±. Tango 37 mai i -35.

$$y=-\frac{72}{8}$$

Whakawehe -72 ki te 8.

$$y=-9$$

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{1}{4} mō te x_{1} me te -9 mō te x_{2}.

$$4y^{2}+35y-9=4\left(y-\frac{1}{4}\right)\left(y-\left(-9\right)\right)$$

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

$$4y^{2}+35y-9=4\left(y-\frac{1}{4}\right)\left(y+9\right)$$

Tango \frac{1}{4} 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.

$$4y^{2}+35y-9=4\times \frac{4y-1}{4}\left(y+9\right)$$

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

$$4y^{2}+35y-9=\left(4y-1\right)\left(y+9\right)$$

Aromātai

$$\left(4y-1\right)\left(y+9\right)$$