Whakaoti i te 2x^2-6x-80=

$$2 x ^ { 2 } - 6 x - 80 =$$

Tauwehe

$$2\left(x-8\right)\left(x+5\right)$$

Ngā Upane Otinga

Tauwehea te 2.

$$2\left(x^{2}-3x-40\right)$$

Whakaarohia te x^{2}-3x-40. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei x^{2}+ax+bx-40. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-3 ab=1\left(-40\right)=-40$$

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 -40.

$$1,-40 2,-20 4,-10 5,-8$$

Tātaihia te tapeke mō ia takirua.

$$1-40=-39 2-20=-18 4-10=-6 5-8=-3$$

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

$$a=-8 b=5$$

Tuhia anō te x^{2}-3x-40 hei \left(x^{2}-8x\right)+\left(5x-40\right).

$$\left(x^{2}-8x\right)+\left(5x-40\right)$$

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

$$x\left(x-8\right)+5\left(x-8\right)$$

Whakatauwehea atu te kīanga pātahi x-8 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-8\right)\left(x+5\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$2\left(x-8\right)\left(x+5\right)$$

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.

$$2x^{2}-6x-80=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 2\left(-80\right)}}{2\times 2}$$

Pūrua -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 2\left(-80\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-6\right)±\sqrt{36-8\left(-80\right)}}{2\times 2}$$

Whakareatia -8 ki te -80.

$$x=\frac{-\left(-6\right)±\sqrt{36+640}}{2\times 2}$$

Tāpiri 36 ki te 640.

$$x=\frac{-\left(-6\right)±\sqrt{676}}{2\times 2}$$

Tuhia te pūtakerua o te 676.

$$x=\frac{-\left(-6\right)±26}{2\times 2}$$

Ko te tauaro o -6 ko 6.

$$x=\frac{6±26}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{6±26}{4}$$

Nā, me whakaoti te whārite x=\frac{6±26}{4} ina he tāpiri te ±. Tāpiri 6 ki te 26.

$$x=\frac{32}{4}$$

Whakawehe 32 ki te 4.

$$x=8$$

Nā, me whakaoti te whārite x=\frac{6±26}{4} ina he tango te ±. Tango 26 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 8 mō te x_{1} me te -5 mō te x_{2}.

$$2x^{2}-6x-80=2\left(x-8\right)\left(x-\left(-5\right)\right)$$

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

$$2x^{2}-6x-80=2\left(x-8\right)\left(x+5\right)$$

Aromātai

$$2\left(x-8\right)\left(x+5\right)$$