Whakaoti i te 8y^2+6y-9

$$8 y ^ { 2 } + 6 y - 9$$

Tauwehe

$$\left(4y-3\right)\left(2y+3\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 8y^{2}+ay+by-9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=6 ab=8\left(-9\right)=-72$$

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 -72.

$$-1,72 -2,36 -3,24 -4,18 -6,12 -8,9$$

Tātaihia te tapeke mō ia takirua.

$$-1+72=71 -2+36=34 -3+24=21 -4+18=14 -6+12=6 -8+9=1$$

Ko te otinga te takirua ka hoatu i te tapeke 6.

$$a=-6 b=12$$

Tuhia anō te 8y^{2}+6y-9 hei \left(8y^{2}-6y\right)+\left(12y-9\right).

$$\left(8y^{2}-6y\right)+\left(12y-9\right)$$

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

$$2y\left(4y-3\right)+3\left(4y-3\right)$$

Whakatauwehea atu te kīanga pātahi 4y-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(4y-3\right)\left(2y+3\right)$$

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.

$$8y^{2}+6y-9=0$$

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.

$$y=\frac{-6±\sqrt{6^{2}-4\times 8\left(-9\right)}}{2\times 8}$$

Pūrua 6.

$$y=\frac{-6±\sqrt{36-4\times 8\left(-9\right)}}{2\times 8}$$

Whakareatia -4 ki te 8.

$$y=\frac{-6±\sqrt{36-32\left(-9\right)}}{2\times 8}$$

Whakareatia -32 ki te -9.

$$y=\frac{-6±\sqrt{36+288}}{2\times 8}$$

Tāpiri 36 ki te 288.

$$y=\frac{-6±\sqrt{324}}{2\times 8}$$

Tuhia te pūtakerua o te 324.

$$y=\frac{-6±18}{2\times 8}$$

Whakareatia 2 ki te 8.

$$y=\frac{-6±18}{16}$$

Nā, me whakaoti te whārite y=\frac{-6±18}{16} ina he tāpiri te ±. Tāpiri -6 ki te 18.

$$y=\frac{12}{16}$$

Whakahekea te hautanga \frac{12}{16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$y=\frac{3}{4}$$

Nā, me whakaoti te whārite y=\frac{-6±18}{16} ina he tango te ±. Tango 18 mai i -6.

$$y=-\frac{24}{16}$$

Whakahekea te hautanga \frac{-24}{16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

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

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{3}{4} mō te x_{1} me te -\frac{3}{2} mō te x_{2}.

$$8y^{2}+6y-9=8\left(y-\frac{3}{4}\right)\left(y-\left(-\frac{3}{2}\right)\right)$$

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

$$8y^{2}+6y-9=8\left(y-\frac{3}{4}\right)\left(y+\frac{3}{2}\right)$$

Tango \frac{3}{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.

$$8y^{2}+6y-9=8\times \frac{4y-3}{4}\left(y+\frac{3}{2}\right)$$

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.

$$8y^{2}+6y-9=8\times \frac{4y-3}{4}\times \frac{2y+3}{2}$$

Whakareatia \frac{4y-3}{4} 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.

$$8y^{2}+6y-9=8\times \frac{\left(4y-3\right)\left(2y+3\right)}{4\times 2}$$

Whakareatia 4 ki te 2.

$$8y^{2}+6y-9=8\times \frac{\left(4y-3\right)\left(2y+3\right)}{8}$$

Whakakorea atu te tauwehe pūnoa nui rawa 8 i roto i te 8 me te 8.

$$8y^{2}+6y-9=\left(4y-3\right)\left(2y+3\right)$$

Aromātai

$$\left(4y-3\right)\left(2y+3\right)$$