Whakaoti i te 5y^2+9y-14

$$5 y ^ { 2 } + 9 y - 14$$

Tauwehe

$$\left(y-1\right)\left(5y+14\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 5y^{2}+ay+by-14. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=9 ab=5\left(-14\right)=-70$$

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 -70.

$$-1,70 -2,35 -5,14 -7,10$$

Tātaihia te tapeke mō ia takirua.

$$-1+70=69 -2+35=33 -5+14=9 -7+10=3$$

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

$$a=-5 b=14$$

Tuhia anō te 5y^{2}+9y-14 hei \left(5y^{2}-5y\right)+\left(14y-14\right).

$$\left(5y^{2}-5y\right)+\left(14y-14\right)$$

Tauwehea te 5y i te tuatahi me te 14 i te rōpū tuarua.

$$5y\left(y-1\right)+14\left(y-1\right)$$

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

$$\left(y-1\right)\left(5y+14\right)$$

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.

$$5y^{2}+9y-14=0$$

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.

$$y=\frac{-9±\sqrt{9^{2}-4\times 5\left(-14\right)}}{2\times 5}$$

Pūrua 9.

$$y=\frac{-9±\sqrt{81-4\times 5\left(-14\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$y=\frac{-9±\sqrt{81-20\left(-14\right)}}{2\times 5}$$

Whakareatia -20 ki te -14.

$$y=\frac{-9±\sqrt{81+280}}{2\times 5}$$

Tāpiri 81 ki te 280.

$$y=\frac{-9±\sqrt{361}}{2\times 5}$$

Tuhia te pūtakerua o te 361.

$$y=\frac{-9±19}{2\times 5}$$

Whakareatia 2 ki te 5.

$$y=\frac{-9±19}{10}$$

Nā, me whakaoti te whārite y=\frac{-9±19}{10} ina he tāpiri te ±. Tāpiri -9 ki te 19.

$$y=\frac{10}{10}$$

Whakawehe 10 ki te 10.

$$y=1$$

Nā, me whakaoti te whārite y=\frac{-9±19}{10} ina he tango te ±. Tango 19 mai i -9.

$$y=-\frac{28}{10}$$

Whakahekea te hautanga \frac{-28}{10} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$y=-\frac{14}{5}$$

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 1 mō te x_{1} me te -\frac{14}{5} mō te x_{2}.

$$5y^{2}+9y-14=5\left(y-1\right)\left(y-\left(-\frac{14}{5}\right)\right)$$

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

$$5y^{2}+9y-14=5\left(y-1\right)\left(y+\frac{14}{5}\right)$$

Tāpiri \frac{14}{5} 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.

$$5y^{2}+9y-14=5\left(y-1\right)\times \frac{5y+14}{5}$$

Whakakorea atu te tauwehe pūnoa nui rawa 5 i roto i te 5 me te 5.

$$5y^{2}+9y-14=\left(y-1\right)\left(5y+14\right)$$

Aromātai

$$\left(y-1\right)\left(5y+14\right)$$