Whakaoti i te y^2-2y-15

$$y ^ { 2 } - 2 y - 15$$

Tauwehe

$$\left(y-5\right)\left(y+3\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei y^{2}+ay+by-15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-2 ab=1\left(-15\right)=-15$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -15.

$$1,-15 3,-5$$

Tātaihia te tapeke mō ia takirua.

$$1-15=-14 3-5=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -2.

$$a=-5 b=3$$

Tuhia anō te y^{2}-2y-15 hei \left(y^{2}-5y\right)+\left(3y-15\right).

$$\left(y^{2}-5y\right)+\left(3y-15\right)$$

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

$$y\left(y-5\right)+3\left(y-5\right)$$

Whakatauwehea atu te kīanga pātahi y-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(y-5\right)\left(y+3\right)$$

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.

$$y^{2}-2y-15=0$$

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.

$$y=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-15\right)}}{2}$$

Pūrua -2.

$$y=\frac{-\left(-2\right)±\sqrt{4-4\left(-15\right)}}{2}$$

Whakareatia -4 ki te -15.

$$y=\frac{-\left(-2\right)±\sqrt{4+60}}{2}$$

Tāpiri 4 ki te 60.

$$y=\frac{-\left(-2\right)±\sqrt{64}}{2}$$

Tuhia te pūtakerua o te 64.

$$y=\frac{-\left(-2\right)±8}{2}$$

Ko te tauaro o -2 ko 2.

$$y=\frac{2±8}{2}$$

Nā, me whakaoti te whārite y=\frac{2±8}{2} ina he tāpiri te ±. Tāpiri 2 ki te 8.

$$y=\frac{10}{2}$$

Whakawehe 10 ki te 2.

$$y=5$$

Nā, me whakaoti te whārite y=\frac{2±8}{2} ina he tango te ±. Tango 8 mai i 2.

$$y=-\frac{6}{2}$$

Whakawehe -6 ki te 2.

$$y=-3$$

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 5 mō te x_{1} me te -3 mō te x_{2}.

$$y^{2}-2y-15=\left(y-5\right)\left(y-\left(-3\right)\right)$$

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

$$y^{2}-2y-15=\left(y-5\right)\left(y+3\right)$$

Aromātai

$$\left(y-5\right)\left(y+3\right)$$