Whakaoti i te x^2-11x-60=

$$x ^ { 2 } - 11 x - 60 =$$

Tauwehe

$$\left(x-15\right)\left(x+4\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei x^{2}+ax+bx-60. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-11 ab=1\left(-60\right)=-60$$

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 -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 -11.

$$a=-15 b=4$$

Tuhia anō te x^{2}-11x-60 hei \left(x^{2}-15x\right)+\left(4x-60\right).

$$\left(x^{2}-15x\right)+\left(4x-60\right)$$

Tauwehea te x i te tuatahi me te 4 i te rōpū tuarua.

$$x\left(x-15\right)+4\left(x-15\right)$$

Whakatauwehea atu te kīanga pātahi x-15 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-15\right)\left(x+4\right)$$

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.

$$x^{2}-11x-60=0$$

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.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-60\right)}}{2}$$

Pūrua -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\left(-60\right)}}{2}$$

Whakareatia -4 ki te -60.

$$x=\frac{-\left(-11\right)±\sqrt{121+240}}{2}$$

Tāpiri 121 ki te 240.

$$x=\frac{-\left(-11\right)±\sqrt{361}}{2}$$

Tuhia te pūtakerua o te 361.

$$x=\frac{-\left(-11\right)±19}{2}$$

Ko te tauaro o -11 ko 11.

$$x=\frac{11±19}{2}$$

Nā, me whakaoti te whārite x=\frac{11±19}{2} ina he tāpiri te ±. Tāpiri 11 ki te 19.

$$x=\frac{30}{2}$$

Whakawehe 30 ki te 2.

$$x=15$$

Nā, me whakaoti te whārite x=\frac{11±19}{2} ina he tango te ±. Tango 19 mai i 11.

$$x=-\frac{8}{2}$$

Whakawehe -8 ki te 2.

$$x=-4$$

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 15 mō te x_{1} me te -4 mō te x_{2}.

$$x^{2}-11x-60=\left(x-15\right)\left(x-\left(-4\right)\right)$$

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

$$x^{2}-11x-60=\left(x-15\right)\left(x+4\right)$$

Aromātai

$$\left(x-15\right)\left(x+4\right)$$