Whakaoti i te p^2-4p-117

$$p ^ { 2 } - 4 p - 117$$

Tauwehe

$$\left(p-13\right)\left(p+9\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei p^{2}+ap+bp-117. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-4 ab=1\left(-117\right)=-117$$

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 -117.

$$1,-117 3,-39 9,-13$$

Tātaihia te tapeke mō ia takirua.

$$1-117=-116 3-39=-36 9-13=-4$$

Ko te otinga te takirua ka hoatu i te tapeke -4.

$$a=-13 b=9$$

Tuhia anō te p^{2}-4p-117 hei \left(p^{2}-13p\right)+\left(9p-117\right).

$$\left(p^{2}-13p\right)+\left(9p-117\right)$$

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

$$p\left(p-13\right)+9\left(p-13\right)$$

Whakatauwehea atu te kīanga pātahi p-13 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(p-13\right)\left(p+9\right)$$

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.

$$p^{2}-4p-117=0$$

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.

$$p=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-117\right)}}{2}$$

Pūrua -4.

$$p=\frac{-\left(-4\right)±\sqrt{16-4\left(-117\right)}}{2}$$

Whakareatia -4 ki te -117.

$$p=\frac{-\left(-4\right)±\sqrt{16+468}}{2}$$

Tāpiri 16 ki te 468.

$$p=\frac{-\left(-4\right)±\sqrt{484}}{2}$$

Tuhia te pūtakerua o te 484.

$$p=\frac{-\left(-4\right)±22}{2}$$

Ko te tauaro o -4 ko 4.

$$p=\frac{4±22}{2}$$

Nā, me whakaoti te whārite p=\frac{4±22}{2} ina he tāpiri te ±. Tāpiri 4 ki te 22.

$$p=\frac{26}{2}$$

Whakawehe 26 ki te 2.

$$p=13$$

Nā, me whakaoti te whārite p=\frac{4±22}{2} ina he tango te ±. Tango 22 mai i 4.

$$p=-\frac{18}{2}$$

Whakawehe -18 ki te 2.

$$p=-9$$

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 13 mō te x_{1} me te -9 mō te x_{2}.

$$p^{2}-4p-117=\left(p-13\right)\left(p-\left(-9\right)\right)$$

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

$$p^{2}-4p-117=\left(p-13\right)\left(p+9\right)$$

Aromātai

$$\left(p-13\right)\left(p+9\right)$$