Whakaoti i te 6x^2-18x-60=

$$6 x ^ { 2 } - 18 x - 60 =$$

Tauwehe

$$6\left(x-5\right)\left(x+2\right)$$

Ngā Upane Otinga

Tauwehea te 6.

$$6\left(x^{2}-3x-10\right)$$

Whakaarohia te x^{2}-3x-10. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei x^{2}+ax+bx-10. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-3 ab=1\left(-10\right)=-10$$

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 -10.

$$1,-10 2,-5$$

Tātaihia te tapeke mō ia takirua.

$$1-10=-9 2-5=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -3.

$$a=-5 b=2$$

Tuhia anō te x^{2}-3x-10 hei \left(x^{2}-5x\right)+\left(2x-10\right).

$$\left(x^{2}-5x\right)+\left(2x-10\right)$$

Tauwehea te x i te tuatahi me te 2 i te rōpū tuarua.

$$x\left(x-5\right)+2\left(x-5\right)$$

Whakatauwehea atu te kīanga pātahi x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-5\right)\left(x+2\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$6\left(x-5\right)\left(x+2\right)$$

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.

$$6x^{2}-18x-60=0$$

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.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 6\left(-60\right)}}{2\times 6}$$

Pūrua -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 6\left(-60\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-\left(-18\right)±\sqrt{324-24\left(-60\right)}}{2\times 6}$$

Whakareatia -24 ki te -60.

$$x=\frac{-\left(-18\right)±\sqrt{324+1440}}{2\times 6}$$

Tāpiri 324 ki te 1440.

$$x=\frac{-\left(-18\right)±\sqrt{1764}}{2\times 6}$$

Tuhia te pūtakerua o te 1764.

$$x=\frac{-\left(-18\right)±42}{2\times 6}$$

Ko te tauaro o -18 ko 18.

$$x=\frac{18±42}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{18±42}{12}$$

Nā, me whakaoti te whārite x=\frac{18±42}{12} ina he tāpiri te ±. Tāpiri 18 ki te 42.

$$x=\frac{60}{12}$$

Whakawehe 60 ki te 12.

$$x=5$$

Nā, me whakaoti te whārite x=\frac{18±42}{12} ina he tango te ±. Tango 42 mai i 18.

$$x=-\frac{24}{12}$$

Whakawehe -24 ki te 12.

$$x=-2$$

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 5 mō te x_{1} me te -2 mō te x_{2}.

$$6x^{2}-18x-60=6\left(x-5\right)\left(x-\left(-2\right)\right)$$

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

$$6x^{2}-18x-60=6\left(x-5\right)\left(x+2\right)$$

Aromātai

$$6\left(x-5\right)\left(x+2\right)$$