Whakaoti i te b^2-17b-60

$$b ^ { 2 } - 17 b - 60$$

Tauwehe

$$\left(b-20\right)\left(b+3\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei b^{2}+pb+qb-60. Hei kimi p me q, whakaritea tētahi pūnaha kia whakaoti.

$$p+q=-17 pq=1\left(-60\right)=-60$$

I te mea kua tōraro te pq, he tauaro ngā tohu o p me q. I te mea kua tōraro te p+q, 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 -60.

$$1,-60 2,-30 3,-20 4,-15 5,-12 6,-10$$

Tātaihia te tapeke mō ia takirua.

$$1-60=-59 2-30=-28 3-20=-17 4-15=-11 5-12=-7 6-10=-4$$

Ko te otinga te takirua ka hoatu i te tapeke -17.

$$p=-20 q=3$$

Tuhia anō te b^{2}-17b-60 hei \left(b^{2}-20b\right)+\left(3b-60\right).

$$\left(b^{2}-20b\right)+\left(3b-60\right)$$

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

$$b\left(b-20\right)+3\left(b-20\right)$$

Whakatauwehea atu te kīanga pātahi b-20 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(b-20\right)\left(b+3\right)$$

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.

$$b^{2}-17b-60=0$$

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.

$$b=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\left(-60\right)}}{2}$$

Pūrua -17.

$$b=\frac{-\left(-17\right)±\sqrt{289-4\left(-60\right)}}{2}$$

Whakareatia -4 ki te -60.

$$b=\frac{-\left(-17\right)±\sqrt{289+240}}{2}$$

Tāpiri 289 ki te 240.

$$b=\frac{-\left(-17\right)±\sqrt{529}}{2}$$

Tuhia te pūtakerua o te 529.

$$b=\frac{-\left(-17\right)±23}{2}$$

Ko te tauaro o -17 ko 17.

$$b=\frac{17±23}{2}$$

Nā, me whakaoti te whārite b=\frac{17±23}{2} ina he tāpiri te ±. Tāpiri 17 ki te 23.

$$b=\frac{40}{2}$$

Whakawehe 40 ki te 2.

$$b=20$$

Nā, me whakaoti te whārite b=\frac{17±23}{2} ina he tango te ±. Tango 23 mai i 17.

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

Whakawehe -6 ki te 2.

$$b=-3$$

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 20 mō te x_{1} me te -3 mō te x_{2}.

$$b^{2}-17b-60=\left(b-20\right)\left(b-\left(-3\right)\right)$$

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

$$b^{2}-17b-60=\left(b-20\right)\left(b+3\right)$$

Aromātai

$$\left(b-20\right)\left(b+3\right)$$