Whakaoti i te 45+4x-x^2

$$45 + 4 x - x ^ { 2 }$$

Tauwehe

$$-\left(x-9\right)\left(x+5\right)$$

Ngā Upane Otinga

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$-x^{2}+4x+45$$

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -x^{2}+ax+bx+45. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=4 ab=-45=-45$$

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 -45.

$$-1,45 -3,15 -5,9$$

Tātaihia te tapeke mō ia takirua.

$$-1+45=44 -3+15=12 -5+9=4$$

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

$$a=9 b=-5$$

Tuhia anō te -x^{2}+4x+45 hei \left(-x^{2}+9x\right)+\left(-5x+45\right).

$$\left(-x^{2}+9x\right)+\left(-5x+45\right)$$

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

$$-x\left(x-9\right)-5\left(x-9\right)$$

Whakatauwehea atu te kīanga pātahi x-9 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-9\right)\left(-x-5\right)$$

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.

$$-x^{2}+4x+45=0$$

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.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-1\right)\times 45}}{2\left(-1\right)}$$

Pūrua 4.

$$x=\frac{-4±\sqrt{16-4\left(-1\right)\times 45}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-4±\sqrt{16+4\times 45}}{2\left(-1\right)}$$

Whakareatia 4 ki te 45.

$$x=\frac{-4±\sqrt{16+180}}{2\left(-1\right)}$$

Tāpiri 16 ki te 180.

$$x=\frac{-4±\sqrt{196}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 196.

$$x=\frac{-4±14}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-4±14}{-2}$$

Nā, me whakaoti te whārite x=\frac{-4±14}{-2} ina he tāpiri te ±. Tāpiri -4 ki te 14.

$$x=\frac{10}{-2}$$

Whakawehe 10 ki te -2.

$$x=-5$$

Nā, me whakaoti te whārite x=\frac{-4±14}{-2} ina he tango te ±. Tango 14 mai i -4.

$$x=-\frac{18}{-2}$$

Whakawehe -18 ki te -2.

$$x=9$$

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 -5 mō te x_{1} me te 9 mō te x_{2}.

$$-x^{2}+4x+45=-\left(x-\left(-5\right)\right)\left(x-9\right)$$

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

$$-x^{2}+4x+45=-\left(x+5\right)\left(x-9\right)$$

Aromātai

$$-\left(x-9\right)\left(x+5\right)$$