Whakaoti i te x^2+x-42

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

Tauwehe

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

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.

$$a+b=1 ab=1\left(-42\right)=-42$$

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 -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.

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

Pūrua 1.

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

Whakareatia -4 ki te -42.

$$x=\frac{-1±\sqrt{1+168}}{2}$$

Tāpiri 1 ki te 168.

$$x=\frac{-1±\sqrt{169}}{2}$$

Tuhia te pūtakerua o te 169.

$$x=\frac{-1±13}{2}$$

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

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

Whakawehe 12 ki te 2.

$$x=6$$

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

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

Whakawehe -14 ki te 2.

$$x=-7$$

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 6 mō te x_{1} me te -7 mō te x_{2}.

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

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

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

Aromātai

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