Whakaoti i te -x^2-x+42

$$- { x }^{ 2 } -x+42$$

Tauwehe

$$\left(6-x\right)\left(x+7\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -x^{2}+ax+bx+42. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=-42=-42$$

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 -42.

$$1,-42 2,-21 3,-14 6,-7$$

Tātaihia te tapeke mō ia takirua.

$$1-42=-41 2-21=-19 3-14=-11 6-7=-1$$

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

$$a=6 b=-7$$

Tuhia anō te -x^{2}-x+42 hei \left(-x^{2}+6x\right)+\left(-7x+42\right).

$$\left(-x^{2}+6x\right)+\left(-7x+42\right)$$

Tauwehea te x i te tuatahi me te 7 i te rōpū tuarua.

$$x\left(-x+6\right)+7\left(-x+6\right)$$

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

$$\left(-x+6\right)\left(x+7\right)$$

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.

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

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.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-1\right)\times 42}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-1\right)±\sqrt{1+4\times 42}}{2\left(-1\right)}$$

Whakareatia 4 ki te 42.

$$x=\frac{-\left(-1\right)±\sqrt{1+168}}{2\left(-1\right)}$$

Tāpiri 1 ki te 168.

$$x=\frac{-\left(-1\right)±\sqrt{169}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 169.

$$x=\frac{-\left(-1\right)±13}{2\left(-1\right)}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±13}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{1±13}{-2}$$

Nā, me whakaoti te whārite x=\frac{1±13}{-2} ina he tāpiri te ±. Tāpiri 1 ki te 13.

$$x=\frac{14}{-2}$$

Whakawehe 14 ki te -2.

$$x=-7$$

Nā, me whakaoti te whārite x=\frac{1±13}{-2} ina he tango te ±. Tango 13 mai i 1.

$$x=-\frac{12}{-2}$$

Whakawehe -12 ki te -2.

$$x=6$$

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 -7 mō te x_{1} me te 6 mō te x_{2}.

$$-x^{2}-x+42=-\left(x-\left(-7\right)\right)\left(x-6\right)$$

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

$$-x^{2}-x+42=-\left(x+7\right)\left(x-6\right)$$

Aromātai

$$\left(6-x\right)\left(x+7\right)$$