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

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

Tauwehe

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

Ngā Upane Otinga

Tauwehea te 2.

$$2\left(-x^{2}+x+30\right)$$

Whakaarohia te -x^{2}+x+30. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -x^{2}+ax+bx+30. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=1 ab=-30=-30$$

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 -30.

$$-1,30 -2,15 -3,10 -5,6$$

Tātaihia te tapeke mō ia takirua.

$$-1+30=29 -2+15=13 -3+10=7 -5+6=1$$

Ko te otinga te takirua ka hoatu i te tapeke 1.

$$a=6 b=-5$$

Tuhia anō te -x^{2}+x+30 hei \left(-x^{2}+6x\right)+\left(-5x+30\right).

$$\left(-x^{2}+6x\right)+\left(-5x+30\right)$$

Tauwehea te -x i te tuatahi me te -5 i te rōpū tuarua.

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

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

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

Me tuhi anō te kīanga whakatauwehe katoa.

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

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.

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

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.

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

Pūrua 2.

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

Whakareatia -4 ki te -2.

$$x=\frac{-2±\sqrt{4+8\times 60}}{2\left(-2\right)}$$

Whakareatia 8 ki te 60.

$$x=\frac{-2±\sqrt{4+480}}{2\left(-2\right)}$$

Tāpiri 4 ki te 480.

$$x=\frac{-2±\sqrt{484}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 484.

$$x=\frac{-2±22}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-2±22}{-4}$$

Nā, me whakaoti te whārite x=\frac{-2±22}{-4} ina he tāpiri te ±. Tāpiri -2 ki te 22.

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

Whakawehe 20 ki te -4.

$$x=-5$$

Nā, me whakaoti te whārite x=\frac{-2±22}{-4} ina he tango te ±. Tango 22 mai i -2.

$$x=-\frac{24}{-4}$$

Whakawehe -24 ki te -4.

$$x=6$$

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 -5 mō te x_{1} me te 6 mō te x_{2}.

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

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

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

Aromātai

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