Whakaoti i te h(x)=-5x^2+20x+60

$$h ( x ) = - 5 x ^ { 2 } + 20 x + 60$$

Tauwehe

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

Ngā Upane Otinga

Tauwehea te 5.

$$5\left(-x^{2}+4x+12\right)$$

Whakaarohia te -x^{2}+4x+12. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -x^{2}+ax+bx+12. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=4 ab=-12=-12$$

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 -12.

$$-1,12 -2,6 -3,4$$

Tātaihia te tapeke mō ia takirua.

$$-1+12=11 -2+6=4 -3+4=1$$

Ko te otinga te takirua ka hoatu i te tapeke 4.

$$a=6 b=-2$$

Tuhia anō te -x^{2}+4x+12 hei \left(-x^{2}+6x\right)+\left(-2x+12\right).

$$\left(-x^{2}+6x\right)+\left(-2x+12\right)$$

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

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

Whakatauwehea atu te kīanga pātahi x-6 mā te whakamahi i te āhuatanga tātai tohatoha.

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

Me tuhi anō te kīanga whakatauwehe katoa.

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

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.

$$-5x^{2}+20x+60=0$$

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.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-5\right)\times 60}}{2\left(-5\right)}$$

Pūrua 20.

$$x=\frac{-20±\sqrt{400-4\left(-5\right)\times 60}}{2\left(-5\right)}$$

Whakareatia -4 ki te -5.

$$x=\frac{-20±\sqrt{400+20\times 60}}{2\left(-5\right)}$$

Whakareatia 20 ki te 60.

$$x=\frac{-20±\sqrt{400+1200}}{2\left(-5\right)}$$

Tāpiri 400 ki te 1200.

$$x=\frac{-20±\sqrt{1600}}{2\left(-5\right)}$$

Tuhia te pūtakerua o te 1600.

$$x=\frac{-20±40}{2\left(-5\right)}$$

Whakareatia 2 ki te -5.

$$x=\frac{-20±40}{-10}$$

Nā, me whakaoti te whārite x=\frac{-20±40}{-10} ina he tāpiri te ±. Tāpiri -20 ki te 40.

$$x=\frac{20}{-10}$$

Whakawehe 20 ki te -10.

$$x=-2$$

Nā, me whakaoti te whārite x=\frac{-20±40}{-10} ina he tango te ±. Tango 40 mai i -20.

$$x=-\frac{60}{-10}$$

Whakawehe -60 ki te -10.

$$x=6$$

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 -2 mō te x_{1} me te 6 mō te x_{2}.

$$-5x^{2}+20x+60=-5\left(x-\left(-2\right)\right)\left(x-6\right)$$

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

$$-5x^{2}+20x+60=-5\left(x+2\right)\left(x-6\right)$$

Aromātai

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