Whakaoti i te x^2-4x-60

$$x ^ { 2 } - 4 x - 60$$

Tauwehe

$$\left(x-10\right)\left(x+6\right)$$

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.

$$a+b=-4 ab=1\left(-60\right)=-60$$

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 -4.

$$a=-10 b=6$$

Tuhia anō te x^{2}-4x-60 hei \left(x^{2}-10x\right)+\left(6x-60\right).

$$\left(x^{2}-10x\right)+\left(6x-60\right)$$

Tauwehea te x i te tuatahi me te 6 i te rōpū tuarua.

$$x\left(x-10\right)+6\left(x-10\right)$$

Whakatauwehea atu te kīanga pātahi x-10 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-10\right)\left(x+6\right)$$

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.

$$x^{2}-4x-60=0$$

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.

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

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-60\right)}}{2}$$

Whakareatia -4 ki te -60.

$$x=\frac{-\left(-4\right)±\sqrt{16+240}}{2}$$

Tāpiri 16 ki te 240.

$$x=\frac{-\left(-4\right)±\sqrt{256}}{2}$$

Tuhia te pūtakerua o te 256.

$$x=\frac{-\left(-4\right)±16}{2}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±16}{2}$$

Nā, me whakaoti te whārite x=\frac{4±16}{2} ina he tāpiri te ±. Tāpiri 4 ki te 16.

$$x=\frac{20}{2}$$

Whakawehe 20 ki te 2.

$$x=10$$

Nā, me whakaoti te whārite x=\frac{4±16}{2} ina he tango te ±. Tango 16 mai i 4.

$$x=-\frac{12}{2}$$

Whakawehe -12 ki te 2.

$$x=-6$$

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 10 mō te x_{1} me te -6 mō te x_{2}.

$$x^{2}-4x-60=\left(x-10\right)\left(x-\left(-6\right)\right)$$

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

$$x^{2}-4x-60=\left(x-10\right)\left(x+6\right)$$

Aromātai

$$\left(x-10\right)\left(x+6\right)$$