Whakaoti i te f(x)=2x^2-8x-42

$$f ( x ) = 2 x ^ { 2 } - 8 x - 42$$

Tauwehe

$$2\left(x-7\right)\left(x+3\right)$$

Ngā Upane Otinga

Tauwehea te 2.

$$2\left(x^{2}-4x-21\right)$$

Whakaarohia te x^{2}-4x-21. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei x^{2}+ax+bx-21. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-4 ab=1\left(-21\right)=-21$$

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 -21.

$$1,-21 3,-7$$

Tātaihia te tapeke mō ia takirua.

$$1-21=-20 3-7=-4$$

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

$$a=-7 b=3$$

Tuhia anō te x^{2}-4x-21 hei \left(x^{2}-7x\right)+\left(3x-21\right).

$$\left(x^{2}-7x\right)+\left(3x-21\right)$$

Tauwehea te x i te tuatahi me te 3 i te rōpū tuarua.

$$x\left(x-7\right)+3\left(x-7\right)$$

Whakatauwehea atu te kīanga pātahi x-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-7\right)\left(x+3\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$2\left(x-7\right)\left(x+3\right)$$

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.

$$2x^{2}-8x-42=0$$

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.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 2\left(-42\right)}}{2\times 2}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 2\left(-42\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-8\right)±\sqrt{64-8\left(-42\right)}}{2\times 2}$$

Whakareatia -8 ki te -42.

$$x=\frac{-\left(-8\right)±\sqrt{64+336}}{2\times 2}$$

Tāpiri 64 ki te 336.

$$x=\frac{-\left(-8\right)±\sqrt{400}}{2\times 2}$$

Tuhia te pūtakerua o te 400.

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

Ko te tauaro o -8 ko 8.

$$x=\frac{8±20}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{8±20}{4}$$

Nā, me whakaoti te whārite x=\frac{8±20}{4} ina he tāpiri te ±. Tāpiri 8 ki te 20.

$$x=\frac{28}{4}$$

Whakawehe 28 ki te 4.

$$x=7$$

Nā, me whakaoti te whārite x=\frac{8±20}{4} ina he tango te ±. Tango 20 mai i 8.

$$x=-\frac{12}{4}$$

Whakawehe -12 ki te 4.

$$x=-3$$

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 7 mō te x_{1} me te -3 mō te x_{2}.

$$2x^{2}-8x-42=2\left(x-7\right)\left(x-\left(-3\right)\right)$$

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

$$2x^{2}-8x-42=2\left(x-7\right)\left(x+3\right)$$

Aromātai

$$2\left(x-7\right)\left(x+3\right)$$