Whakaoti i te y^2+13y-68

$$y ^ { 2 } + 13 y - 68$$

Tauwehe

$$\left(y-4\right)\left(y+17\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei y^{2}+ay+by-68. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=13 ab=1\left(-68\right)=-68$$

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 -68.

$$-1,68 -2,34 -4,17$$

Tātaihia te tapeke mō ia takirua.

$$-1+68=67 -2+34=32 -4+17=13$$

Ko te otinga te takirua ka hoatu i te tapeke 13.

$$a=-4 b=17$$

Tuhia anō te y^{2}+13y-68 hei \left(y^{2}-4y\right)+\left(17y-68\right).

$$\left(y^{2}-4y\right)+\left(17y-68\right)$$

Tauwehea te y i te tuatahi me te 17 i te rōpū tuarua.

$$y\left(y-4\right)+17\left(y-4\right)$$

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

$$\left(y-4\right)\left(y+17\right)$$

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.

$$y^{2}+13y-68=0$$

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.

$$y=\frac{-13±\sqrt{13^{2}-4\left(-68\right)}}{2}$$

Pūrua 13.

$$y=\frac{-13±\sqrt{169-4\left(-68\right)}}{2}$$

Whakareatia -4 ki te -68.

$$y=\frac{-13±\sqrt{169+272}}{2}$$

Tāpiri 169 ki te 272.

$$y=\frac{-13±\sqrt{441}}{2}$$

Tuhia te pūtakerua o te 441.

$$y=\frac{-13±21}{2}$$

Nā, me whakaoti te whārite y=\frac{-13±21}{2} ina he tāpiri te ±. Tāpiri -13 ki te 21.

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

Whakawehe 8 ki te 2.

$$y=4$$

Nā, me whakaoti te whārite y=\frac{-13±21}{2} ina he tango te ±. Tango 21 mai i -13.

$$y=-\frac{34}{2}$$

Whakawehe -34 ki te 2.

$$y=-17$$

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 4 mō te x_{1} me te -17 mō te x_{2}.

$$y^{2}+13y-68=\left(y-4\right)\left(y-\left(-17\right)\right)$$

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

$$y^{2}+13y-68=\left(y-4\right)\left(y+17\right)$$

Aromātai

$$\left(y-4\right)\left(y+17\right)$$