Whakaoti i te 30+7x-x^2

$$30 + 7 x - x ^ { 2 }$$

Tauwehe

$$-\left(x-10\right)\left(x+3\right)$$

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.

$$-x^{2}+7x+30$$

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -x^{2}+ax+bx+30. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=7 ab=-30=-30$$

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 -30.

$$-1,30 -2,15 -3,10 -5,6$$

Tātaihia te tapeke mō ia takirua.

$$-1+30=29 -2+15=13 -3+10=7 -5+6=1$$

Ko te otinga te takirua ka hoatu i te tapeke 7.

$$a=10 b=-3$$

Tuhia anō te -x^{2}+7x+30 hei \left(-x^{2}+10x\right)+\left(-3x+30\right).

$$\left(-x^{2}+10x\right)+\left(-3x+30\right)$$

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

$$-x\left(x-10\right)-3\left(x-10\right)$$

Whakatauwehea atu te kīanga pātahi x-10 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-10\right)\left(-x-3\right)$$

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.

$$-x^{2}+7x+30=0$$

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.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-1\right)\times 30}}{2\left(-1\right)}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\left(-1\right)\times 30}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-7±\sqrt{49+4\times 30}}{2\left(-1\right)}$$

Whakareatia 4 ki te 30.

$$x=\frac{-7±\sqrt{49+120}}{2\left(-1\right)}$$

Tāpiri 49 ki te 120.

$$x=\frac{-7±\sqrt{169}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 169.

$$x=\frac{-7±13}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-7±13}{-2}$$

Nā, me whakaoti te whārite x=\frac{-7±13}{-2} ina he tāpiri te ±. Tāpiri -7 ki te 13.

$$x=\frac{6}{-2}$$

Whakawehe 6 ki te -2.

$$x=-3$$

Nā, me whakaoti te whārite x=\frac{-7±13}{-2} ina he tango te ±. Tango 13 mai i -7.

$$x=-\frac{20}{-2}$$

Whakawehe -20 ki te -2.

$$x=10$$

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 -3 mō te x_{1} me te 10 mō te x_{2}.

$$-x^{2}+7x+30=-\left(x-\left(-3\right)\right)\left(x-10\right)$$

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

$$-x^{2}+7x+30=-\left(x+3\right)\left(x-10\right)$$

Aromātai

$$-\left(x-10\right)\left(x+3\right)$$