Whakaoti i te f(x)=6x-2x^2+20

$$f ( x ) = 6 x - 2 x ^ { 2 } + 20$$

Tauwehe

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

Ngā Upane Otinga

Tauwehea te 2.

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

Whakaarohia te 3x-x^{2}+10. Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$-x^{2}+3x+10$$

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.

$$a+b=3 ab=-10=-10$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. 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.

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

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.

$$-2x^{2}+6x+20=0$$

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.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-2\right)\times 20}}{2\left(-2\right)}$$

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\left(-2\right)\times 20}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-6±\sqrt{36+8\times 20}}{2\left(-2\right)}$$

Whakareatia 8 ki te 20.

$$x=\frac{-6±\sqrt{36+160}}{2\left(-2\right)}$$

Tāpiri 36 ki te 160.

$$x=\frac{-6±\sqrt{196}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 196.

$$x=\frac{-6±14}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-6±14}{-4}$$

Nā, me whakaoti te whārite x=\frac{-6±14}{-4} ina he tāpiri te ±. Tāpiri -6 ki te 14.

$$x=\frac{8}{-4}$$

Whakawehe 8 ki te -4.

$$x=-2$$

Nā, me whakaoti te whārite x=\frac{-6±14}{-4} ina he tango te ±. Tango 14 mai i -6.

$$x=-\frac{20}{-4}$$

Whakawehe -20 ki te -4.

$$x=5$$

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 -2 mō te x_{1} me te 5 mō te x_{2}.

$$-2x^{2}+6x+20=-2\left(x-\left(-2\right)\right)\left(x-5\right)$$

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

$$-2x^{2}+6x+20=-2\left(x+2\right)\left(x-5\right)$$

Aromātai

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