Whakaoti i te -x^2-3x+28=

$$- x ^ { 2 } - 3 x + 28 =$$

Tauwehe

$$\left(4-x\right)\left(x+7\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -x^{2}+ax+bx+28. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-3 ab=-28=-28$$

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 -28.

$$1,-28 2,-14 4,-7$$

Tātaihia te tapeke mō ia takirua.

$$1-28=-27 2-14=-12 4-7=-3$$

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

$$a=4 b=-7$$

Tuhia anō te -x^{2}-3x+28 hei \left(-x^{2}+4x\right)+\left(-7x+28\right).

$$\left(-x^{2}+4x\right)+\left(-7x+28\right)$$

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

$$x\left(-x+4\right)+7\left(-x+4\right)$$

Whakatauwehea atu te kīanga pātahi -x+4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-x+4\right)\left(x+7\right)$$

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.

$$-x^{2}-3x+28=0$$

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.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-1\right)\times 28}}{2\left(-1\right)}$$

Pūrua -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-1\right)\times 28}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-3\right)±\sqrt{9+4\times 28}}{2\left(-1\right)}$$

Whakareatia 4 ki te 28.

$$x=\frac{-\left(-3\right)±\sqrt{9+112}}{2\left(-1\right)}$$

Tāpiri 9 ki te 112.

$$x=\frac{-\left(-3\right)±\sqrt{121}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 121.

$$x=\frac{-\left(-3\right)±11}{2\left(-1\right)}$$

Ko te tauaro o -3 ko 3.

$$x=\frac{3±11}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{3±11}{-2}$$

Nā, me whakaoti te whārite x=\frac{3±11}{-2} ina he tāpiri te ±. Tāpiri 3 ki te 11.

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

Whakawehe 14 ki te -2.

$$x=-7$$

Nā, me whakaoti te whārite x=\frac{3±11}{-2} ina he tango te ±. Tango 11 mai i 3.

$$x=-\frac{8}{-2}$$

Whakawehe -8 ki te -2.

$$x=4$$

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 4 mō te x_{2}.

$$-x^{2}-3x+28=-\left(x-\left(-7\right)\right)\left(x-4\right)$$

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

$$-x^{2}-3x+28=-\left(x+7\right)\left(x-4\right)$$

Aromātai

$$\left(4-x\right)\left(x+7\right)$$