Whakaoti i te 9y^2+75y-54

$$9 y ^ { 2 } + 75 y - 54$$

Tauwehe

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

Ngā Upane Otinga

Tauwehea te 3.

$$3\left(3y^{2}+25y-18\right)$$

Whakaarohia te 3y^{2}+25y-18. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 3y^{2}+ay+by-18. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=25 ab=3\left(-18\right)=-54$$

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 -54.

$$-1,54 -2,27 -3,18 -6,9$$

Tātaihia te tapeke mō ia takirua.

$$-1+54=53 -2+27=25 -3+18=15 -6+9=3$$

Ko te otinga te takirua ka hoatu i te tapeke 25.

$$a=-2 b=27$$

Tuhia anō te 3y^{2}+25y-18 hei \left(3y^{2}-2y\right)+\left(27y-18\right).

$$\left(3y^{2}-2y\right)+\left(27y-18\right)$$

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

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

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

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

Me tuhi anō te kīanga whakatauwehe katoa.

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

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.

$$9y^{2}+75y-54=0$$

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.

$$y=\frac{-75±\sqrt{75^{2}-4\times 9\left(-54\right)}}{2\times 9}$$

Pūrua 75.

$$y=\frac{-75±\sqrt{5625-4\times 9\left(-54\right)}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$y=\frac{-75±\sqrt{5625-36\left(-54\right)}}{2\times 9}$$

Whakareatia -36 ki te -54.

$$y=\frac{-75±\sqrt{5625+1944}}{2\times 9}$$

Tāpiri 5625 ki te 1944.

$$y=\frac{-75±\sqrt{7569}}{2\times 9}$$

Tuhia te pūtakerua o te 7569.

$$y=\frac{-75±87}{2\times 9}$$

Whakareatia 2 ki te 9.

$$y=\frac{-75±87}{18}$$

Nā, me whakaoti te whārite y=\frac{-75±87}{18} ina he tāpiri te ±. Tāpiri -75 ki te 87.

$$y=\frac{12}{18}$$

Whakahekea te hautanga \frac{12}{18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

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

Nā, me whakaoti te whārite y=\frac{-75±87}{18} ina he tango te ±. Tango 87 mai i -75.

$$y=-\frac{162}{18}$$

Whakawehe -162 ki te 18.

$$y=-9$$

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{2}{3} mō te x_{1} me te -9 mō te x_{2}.

$$9y^{2}+75y-54=9\left(y-\frac{2}{3}\right)\left(y-\left(-9\right)\right)$$

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

$$9y^{2}+75y-54=9\left(y-\frac{2}{3}\right)\left(y+9\right)$$

Tango \frac{2}{3} 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.

$$9y^{2}+75y-54=9\times \frac{3y-2}{3}\left(y+9\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 3 i roto i te 9 me te 3.

$$9y^{2}+75y-54=3\left(3y-2\right)\left(y+9\right)$$

Aromātai

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